Tuesday, July 2, 2019

Irreducibility of cyclotomic polynomials

For every integer $n\geq 1$, the $n$th cyclotomic polynomial $\Phi_n$ is the monic polynomial whose complex roots are the primitive $n$th roots of unity. A priori, this is a polynomial with complex coefficients, but since every $n$th root of unity is a primitive $d$th root of unity, for a unique divisor $d$ of $n$, one has the relation
\[ T^n-1 = \prod_{d\mid n} \Phi_d(T), \]
which implies, by induction and euclidean divisions, that $\Phi_n \in \mathbf Z[T]$ for every $n$.
The degree of the polynomial $\Phi_n$ is $\phi(n)$, the Euler indicator, number of units in $\mathbf Z/n\mathbf Z$, or number of integers in $\{0,1,\dots,n-1\}$ which are prime to $n$.

The goal of this note is to explain a few proofs that these polynomials are irreducible in $\mathbf Q[T]$ — or equivalently, in view of Gauss's lemma, in $\mathbf Z[T]$. This also amounts to saying that $\deg(\Phi_n)=\phi(n)$ or that the cyclotomic extension has degree $\phi(n)$, or that the canonical group homomorphism from the Galois group of $\mathbf Q(\zeta_n)$ to $(\mathbf Z/n\mathbf Z)^\times$ is an isomorphism.

1. The case where $n=p$ is a prime number.

One has $T^p-1=(T-1)(T^{p+1}+\dots+1)$, hence $\Phi_p=T^{p-1}+\dots+1$. If one reduces it modulo $p$, one finds $\Phi_p(T)\equiv (T-1)^{p-1}$, because $(T-1)\Phi_p(T)=T^p-1\equiv (T-1)^p$. Moreover, $\Phi_p(1)=p$ is not a multiple of $p^2$. By the Eisenstein criterion (after a change of variables $T=1+U$, if one prefers), the polynomial $\Phi_p$ is irreducible.

This argument also works when $n=p^e$ is a power of a prime number. Indeed, since a complex number $\alpha$ is a primitive $p^e$th root of unity if and only if $\alpha^{p^{e-1}}$ is a primitive $p$th root of unity, one has $\Phi_{p^e}= \Phi_p(T^{p^{e-1}})$. Then the Eisenstein criterion gives the result.

Comment.From the point of view of algebraic number theory, this proof makes use of the fact that the cyclotomic extension $\mathbf Q(\zeta_p)$ is totally ramified at $p$, of ramification index $p-1$.
Consequently, it must have degree $p-1$. More generally, it will prove that $\Phi_p$ is irreducible over the field $\mathbf Q_p$ of $p$-adic numbers, or even over any unramified extension of it, or even over any algebraic extension of $\mathbf Q_p$ for which the ramification index is prime to $p-1$.


2. The classical proof

Let us explain a proof that works for all integer $n$. Let $\alpha$ be a primitive $n$th root of unity, and let $P\in\mathbf Z[T]$ be its minimal polynomial — one has $P\mid \Phi_n$ in $\mathbf Z[T]$. Let (A priori, the divisibility is in $\mathbf Q[T]$, but Gauss's lemma implies that it holds in $\mathbf Z[T]$ as well.) Fix a polynomial $Q\in\mathbf Z[T]$ such that $\Phi_n=PQ$.

By euclidean division, one sees that the set $\mathbf Z[\alpha]$ of complex numbers of the form $S(\alpha)$, for $S\in\mathbf Z[T]$, is a free abelian group of rank $\deg(P)$, with basis $1,\alpha,\dots,\alpha^{\deg(P)-1}$.

Let $p$ be a prime number which does not divide $n$. By Fermat's little theorem, one has $P(T)^p \equiv P(T^p) \pmod p$, so that there exists $P_1\in\mathbf Z[T]$ such that $P(X)^p-P(X^p)=pP_1(T)$. This implies that $P(\alpha^p)=p P_1(\alpha)\in p\mathbf Z[\alpha]$.

Since $p$ is prime to $n$, $\alpha^p$ is a primitive $n$th root of unity, hence $\Phi_n(\alpha^p)=0$. Assume that $P(\alpha^p)\neq 0$. Then one has $Q(\alpha^p)=0$. Differentiating the equality $\Phi_n=PQ$, one gets $nT^{n}=T\Phi'_n(T)=TP'Q+TPQ'$; let us evaluate this at $\alpha_p$, we obtain $n=\alpha^p P(\alpha_p) Q'(\alpha^p)=p \alpha^p P^1(\alpha^p)Q'(\alpha^p)$. In other words, $n\in p\mathbf Z[\alpha]$, which is absurd because $n$ does not divide $p$. Consequently, $P(\alpha^p)=0$, and $P$ is also the minimal polynomial of $\alpha^p$.

By induction, one has $P(\alpha^m)=0$ for every integer $m$ which is prime to $n$. All primitive $n$th roots of unity are roots of $P$ and $\deg(P)=\phi(n)=\deg(\Phi_n)$. This shows that $P=\Phi_n$.

Comment.Since this proof considers prime numbers $p$ which do not divide $n$, it makes implicit use of the fact that the cyclotomic extension is unramified away from primes dividing $n$. The differentiation that appears in the proof is a way of proving this non-ramification: if $P(\alpha^p)$ is zero modulo $p$, it must be zero.

3. Landau's proof

A 1929 paper by Landau gives a variant of this classical proof which I just learnt from Milne's notes on Galois theory and which I find significantly easier.

We start as previously, $\alpha$ being a primitive $n$th root of unity and $P\in\mathbf Z[T]$ being its minimal polynomial.

Let us consider, when $k$ varies, the elements $P(\alpha^k)$ of $\mathbf Z[\alpha]$. There are finitely many of them, since this sequence is $n$-periodic, so that they can be written as finitely polynomials of degree $<\deg(P)$ in $\alpha$. Let $A$ be an upper-bound for their coefficients. If $p$ is a prime number, we have $P(\alpha^p) \in p\mathbf Z[\alpha]$ (by an already given argument). This implies $P(\alpha^p)=0$ if $p>A$.

By induction, one has $P(\alpha^m)=0$ for any integer $m$ whose prime factors are all $>A$.

One the other hand, if $m$ is an integer prime to $n$ and $P$ is the product of all prime number $p$ such that $p\leq A$ and $p$ does not divide $m$, then $m'=m+nP $ is another integer all of which prime factors are $>A$. (Indeed, if $p\leq A$, then either $p\mid m$ in
which case $p\nmid nP$ so that then $p\nmid m'$, or $p\nmid m$ in which case $p\mid nP$ so that $p\nmid m'$.) Since $m'\equiv m \pmod n$, one has $P(\alpha^{m})=P(\alpha^{m'})=0$.

This shows that all primitive $n$th roots of unity are roots of $P$, hence $P=\Phi_n$.

Comment. —This proof is quite of a mysterious nature to me.

4. Using Galois theory to pass from local information to global information

The cyclotomic extension $K_n$ contains, as subextension, the cyclotomic extensions $K_{p^e}$, where $n=\prod p_i^{e_i}$ is the decomposition of $n$ has a product of powers of prime numbers. By the first case, $K_{p^e}$ has degree $\phi(p^e)=p^{e-1}(p-1)$ over $\mathbf Q$. To prove that $\Phi_n$ is irreducible, it suffices to prove that these extensions are linearly disjoint, which is the object of the following lemma.

Lemma. — Let $m$ and $n$ be integers and let $d$ be their gcd. Then $K_m\cap K_n=K_d$.

This is an application of Galois theory (and the result holds for every ground field as soon as its characteristic does not divide $m$ and $n$). Let $M$ be the least common multiple of $m$ and $n$. One has $K_N=K_m\cdot K_n$, and the cyclotomic character furnishes a group morphism $\operatorname{Gal}(K_N/\mathbf Q)\to (\mathbf Z/N\mathbf Z)^\times$. The Galois groups $\operatorname{Gal}(K_N/K_m)$ and $\operatorname{Gal}(K_N/K_n)$ corresponding to the subfields $K_m$ and $K_n$ are the kernels of the composition of the cyclotomic character with the projections to $(\mathbf Z/m\mathbf Z)^\times$ and $(\mathbf Z/n\mathbf Z)^\times$, and their intersection to the subgroup generated by these two kernels, which is none but the kernel of the composition of the cyclotomic character with the projection to $(\mathbf Z/d\mathbf Z)^\times$.

Sunday, May 19, 2019

Designs, Skolem sequences, and partitions of integers

Recently, my father offered me the first volume of a graphic novel by Jean-François Kierzkowski and Marek called La suite de Skolem — Skolem's sequences. I knew about the norwegian mathematician Thoralf Skolem for two different reasons (the Löwenheim-Skolem theorem in model theory, and some diophantine equations that Laurent Moret-Bailly put in a more geometric setting — see his series of papers on Problèmes de Skolem), but I had never heared about Skolem sequences.

They appear in his 1957 paper, On certain distributions of integers in pairs with given differences (Math Scand., 5, 57-68).
The question is to put the integers $1,2,\dots,2n$ in $n$ pairs $(a_1,b_1),\dots,(a_n,b_n)$ such that the differences are all different, namely $b_i-a_i=i$ for $i\in\{1,\dots,r\}$. One can put it differently: write a sequence of $2n$ integers, where each of the integers from $1$ to $n$ appear exactly twice, the two $1$s being at distance $1$, the two $2$s at distance $2$, etc.
For example, $4,2,3,2,4,3,1,1$ is a Skolem sequence of length $n$, corresponding to the pairs $(7,8), (2,4), (3,6),(1,5)$.

Le jeu des cavaliers (Jessica Stockholder) — photo V. Pantaloni
Le jeu des cavaliers (photo V. Pantaloni)
The possibility of such sequences has been materialized under the form of a giant sculpture Le jeu des cavaliers by Jessica Stockholder at the Institut des Hautes Études Scientifiques (IHÉS) at Bures sur Yvette.

There is a basic necessary and sufficient condition for such a sequence to exist, namely $n$ has to be congruent to $0$ or $1$ modulo $4$. The proof of necessity is easy (attributed by Skolem to Bang): one has $\sum_{i=1}^n(b_i-a_i)=n(n+1)/2$, and $\sum_{i=1}^n (b_i+a_i)=2n(2n+1)/2$, so that $\sum_{i=1}^n b_i=n(n+1)/4+2n(2n+1)/2=n(5n+3)/4$. If $n$ is even, this forces $n\equiv 0 \pmod 4$, while if $n$ is odd, this forces $5n+3\equiv 0\pmod 4$, hence $n\equiv 1\pmod 4$. The proof of  existence consists in an explicit example of such a sequence, which is written down in Skolem's paper.

Skolem's motivation is only alluded to in that paper, but he explains it a bit further next year. In Some Remarks on the Triple Systems of Steiner, he gives the recipe that furnishes such a system from a Skolem sequence. Steiner triple systems on a set $S$ is the datum of triplets of elements of $S$ such that each pair of two elements of $S$ appears exactly once. In other words, they are a $(3,2,1)$-design on $S$ — a $(m,p,q)$-design on a set $S$ being a family of $m$-subsets of $S$ such that each $q$-subset appears in exactly $p$ of those subsets. Some relatively obvious divisibility conditions can be written down that give a necessary condition for the existence of designs with given parameters, but actual existence is much more difficult. In fact, it has been shown only recently by Peter Keevash that these necessary conditions are sufficient, provided the cardinality of the set $S$ is large enough, see Gil Kalai's talk Designs exist! at the Bourbaki Seminar.

In the case of Steiner triple systems, the condition is that the number $s$ of elements of $S$ be congruent to $1$ or $3$ modulo $6$. Indeed, there are $s(s-1)/2$ pairs of elements of $S$, and each 3-subset of the triple system features 3 such pairs, so that there are $N=s(s-1)/6$ triples. On the other hand, each element of $S$ appears exactly $3N/s$ times, so that $(s-1)/2$ is an integer. So $s$ has to be odd, and either $3$ divides $s$ (in which case $s\equiv 3\pmod 6$) or $s\equiv 1\pmod 6$.

And Skolem's observation is that a family of $n$ pairs $(a_i,b_i)$ as above furnishes a triple system on the set $S=\mathbf Z/(6n+1)\mathbf Z$, namely the triples $(i,i+j,i+b_j+n)$ where $1\leq i,j\leq n$, thus constructing Steiner triple systems on a set whose cardinality $6n+1$, when $n\equiv 0,1\pmod 4$.

My surprise came at the reading of the rest of Skolem's 1957 paper, because I knew the result he then described but had no idea it was due to him. (In fact, it was one of the first homework my math teacher Johan Yebbou gave to us when I was in classes préparatoires.) And since this result is very nice, let me tell you about it.

Theorem.Let $\alpha>1$ and $\beta>1$ be irrational real numbers such that $\alpha^{-1}+\beta^{-1}=1$. Then each strictly positive integer can be written either as $\lfloor \alpha n\rfloor$, or $\lfloor \beta n\rfloor$ for some integer $n\geq 1$, but not of both forms.

First of all, assume $N=\lfloor \alpha n\rfloor=\lfloor \beta m\rfloor$. Using that $\alpha,\beta$ are irrational, we thus write $N< \alpha n<N+1$ and $N<\beta m<N+1$. Dividing these inequalities by $\alpha$ and $\beta$ and adding them, we get $N<n+m<N+1$, since $\alpha^{-1}+\beta^{-1}=1$. This proves that any given integer can be written only of one of those two forms.

Since $\alpha^{-1}+\beta^{-1}=1$, one of $\alpha,\beta$ has to be $<2$. Assume that $1<\alpha<2$. The integers of the form $\lfloor \alpha n\rfloor$ form a strictly increasing sequence, and we want to show that any integer it avoids can be written $\lfloor \beta m\rfloor$.

Set $\gamma=\alpha-1$, so that $\beta=\alpha/(\alpha-1)=1+1/\gamma$. 

For every integer, we have $\lfloor \alpha(n+1)\rfloor = \lfloor \alpha n\rfloor + 1$ or $\lfloor \alpha(n+1)\rfloor=\lfloor \alpha n\rfloor+2$, so that if $\lfloor \alpha n\rfloor + 1$ is avoided, one has $\lfloor \alpha (n+1)\rfloor=\lfloor\alpha n\rfloor +2$.

Then, $\lfloor \alpha n\rfloor = n+\lfloor \gamma n\rfloor=n+k-1$, where $k=1+\lfloor \gamma n\rfloor$. The inequalities $k-1<\gamma n <k$ imply $k/\gamma - 1/\gamma< n<k/\gamma$. Moreover, $\lfloor \alpha(n+1)\rfloor=n+1+\lfloor \gamma(n+1)\rfloor=n+k+1$, so that $k+1<\gamma(n+1)<k+2$, hence $n>k/\gamma+1/\gamma-1>k/\gamma-1$. This proves that $n=\lfloor k/\gamma\rfloor$. Then, $\lfloor k\beta\rfloor=k+\lfloor k/\gamma\rfloor=k+n=\lfloor \alpha n\rfloor +1$.






Saturday, May 12, 2018

A theorem of Lee—Yang on root of polynomials

A recent MathOverflow post asked for a proof that the roots of certain polynomials were located on the unit circle. A comment by Richard Stanley pointed to a beautiful theorem of T. D. Lee and C. N. Yang. By the way, these two authors were physicists, got the Nobel prize in physics in 1957, and were the first Chinese scientists to be honored by the Nobel prize.

This theorem appears in an appendix to their paper, Statistical Theory of Equations of State and Phase Transitions.IL Lattice Gas and Ising Model, published in Phys. Review in 1952, and devoted to properties of the partition function of some lattice gases. Here is a discussion of this theorem, following both the initial paper and notes by Shayan Oveis Gharan.

Let $A=(a_{i,j})_{1\leq i,j\leq n}$ be a Hermitian matrix ($a_{i,j}=\overline{a_{j,i}}$ for all $i,j$). Define a polynomial
\[ F(T) = \sum_{S\subset\{1,\dots,n\}} \prod_{\substack{i\in S \\ j\not\in S}} a_{i,j} T^{\# S}. \]

Theorem 1 (Lee-Yang). — If $\lvert a_{i,j}\rvert\leq 1$ for all $i,j$, then all roots of $F$ have absolute value $1$.

This theorem follows from a multivariate result. Let us define
\[ P(T_1,\dots,T_n) = \sum_{S\subset\{1,\dots,n\}} \prod_{\substack{i\in S \\ j\not\in S}} a_{i,j} \prod_{i\in S} T_i .\]
Say that a polynomial $F\in\mathbf C[T_1,\dots,T_n]$ is good if it has no root $(z_1,\dots,z_n)\in\mathbf C^n$ such that $\lvert z_i\rvert <1$ for all $i$.

Proposition 2 (Lee-Yang). — If $\lvert a_{i,j}\rvert\leq 1$ for all $i,j$, then $P$ is good.

For every pair $(i,j)$, set $a^S_{i,j}=a_{i,j}$ if $i$ belongs to $S$, but not $j$, and set $a^S_{i,j}=1$ otherwise. Consequently,
\[ P(T_1,\dots,T_n) = \sum_{S\subset\{1,\dots,n\}}\prod_{i,j} a^S_{i,j} \prod_{k\in S} T_k. \]
In other words, if we define polynomials
\[ P_{i,j} (T_1,\dots,T_n) = \sum_{S\subset\{1,\dots,n\}} a^S_{i,j} \prod_{k\in S} T_k, \]
then $P$ is the “coefficientwise product” of the polynomials $P_{i,j}$.
We also note that these polynomials have degree at most one with respect to every variable. These observations may motivate the following lemmas concerning good polynomials.

Lemma 1. — If $P,Q$ are good, then so is their product.

Lemma 2. — If $P(T_1,\dots,T_n)$ is good, then $P(a,T_2,\dots,T_n)$ is good for every $a\in\mathbf C$ such that $\lvert a\rvert \leq 1$.

This is obvious if $\lvert a\rvert <1$; in the general case, this follows from the Rouché theorem — the set polynomials (of bounded degree) whose roots belong to some closed subset is closed.

Lemma 3. — If $\lvert a\rvert \leq 1$, then $1+aT$ is good.

This is obvious.

Lemma 4. — If $\lvert a\rvert\leq 1$, then $P=1+aT_1+\bar a T_2+ T_1T_2$ is good.

If $\lvert a\rvert =1$, then $P=(1+aT_1)(1+\bar aT_2)$ is good, as the product of two good polynomials.
Now assume that $\lvert a\rvert <1$. Let $(z_1,z_2)$ be a root of $P$ such that $\lvert z_1\rvert<1$. One has
\[ z_2 = - \frac{1+az_1}{z_1+\bar a}. \]
Since the Möbius transformation $z\mapsto (z+\bar a)/(1+a z)$ defines a bijection from the unit open disk to itself, one has $\lvert z_2\rvert >1$.

Lemma 5. — If $P=a+bT_1+cT_2+dT_1T_2$ is good, then $Q=a+dT$ is good.

Assume otherwise, so that $\lvert a\rvert <\lvert d\rvert$. By symmetry, we assume $\lvert b\rvert\geq \lvert c\rvert$. We write $P (T_1,T_2) = (a+cT_2) + (b+dT_2) T_1$.
Choose $z_2\in\mathbf C$ such that $dz_2$ and $b$ have the same argument; if, moreover $z_2$ is close enough to $1$ and satisfies $\lvert z_2\rvert <1$, then
\[ \lvert b+dz_2\rvert=\lvert b\rvert +\lvert dz_2\rvert > \lvert a\rvert+\lvert c\rvert>\lvert a+cz_2\rvert. \]
Consequently, the polynomial $P(T_1,z_2)$ is not good; a contradiction.

Lemma 6. — If $P,Q$ are good polynomials of degree at most one in each variable, then so is their coefficientwise product.

We first treat the case of one variable: then $P=a+bT$ and $Q=a'+b'T$, so that their coefficientwise product is given by $R=aa'+bb'T$. By assumption $\lvert a\rvert \geq \lvert b\rvert$ and $a\neq 0$;
similarly, $\lvert a'\rvert \geq \lvert b'\rvert $ and $a'\neq 0$. Consequently, $aa'\neq0$ and $\lvert aa'\rvert \geq \lvert bb'\rvert$, which shows that $R$ is good.
We prove the result by induction on $n$. For every subset $S$ of $\{1,\dots,n-1\}$, let $a_S$ and $b_S$ be the coefficients of $\prod_{i\in S}T_i$ and of $\prod_{i\in S} T_i \cdot T_n$ in $P$; define similarly $c_S$ and $d_S$with $Q$. The coefficientwise product of $P$ and $Q$ is equal to
\[ R= \sum_S (a_S c_S +b_S d_S T_n ) \prod_{i\in S} T_i . \]
Let $z\in\mathbf C$ be such that $\lvert z\rvert \leq 1$, so that
\[ P(T_1,\dots,T_{n-1},z)= \sum _S (a_S+b_S z) \prod_{i\in S} T_i \] is good, by lemma 2. Similarly, for $w\in\mathbf C$ such that $\lvert w\rvert\leq 1$, $Q(T_1,\dots,T_{n-1},w)
=\sum _S (c_S+d_S w) \prod_{i\in S} T_i$ is good. By induction, their coefficientwise product, given by
\[ R_{z,w} = \sum_S (a_S+b_S z)(c_S+d_S w) \prod_{i\in S} T_i \]
is good as well.
We now fix complex numbers $z_1,\dots,z_{n-1}$ of absolute value $<1$. By what precedes, the polynomial
\[ S(T,U) = (\sum_S a_S c_S z_S) + (\sum_S b_Sc_S z_S) T + (\sum_S a_S d_S z_S) U
+ (\sum_S b_S d_S z_S) TU \]
is good, where $z_S=\prod_{i\in S}z_i$. According to lemma 4, the polynomial
\[ R(z_1,\dots,z_{n-1},T) = (\sum_S a_S c_S z_S) + (\sum_S b_S d_S z_S) T \]
is good. This proves that $R$ is good.

Proof of theorem 2. — We have already observed that the polynomial $P$ is the coefficientwise product of polynomials $P_{i,j}$, each of them has degree at most one in each variable. On the other hand, one has
\[ P_{i,j} = (1+a_{i,j} T_i + a_{j,i} T_j + T_i T_j) \prod_{k\neq i,j} (1+T_k), \]
a product of good polynomials, so that $P_{i,j}$ is good. This proves that $P$ is good.

In fact, more is true. Indeed, one has
\[
\begin{align*} T_1\dots T_n P(1/T_1,\dots,1/T_n)
& = \sum_ S \prod_{\substack{i\in S \\ j\notin S}} a_{i,j} \prod_{i\notin S} T_i \\
& =P^*(T_1,\dots,T_n)
\end{align*}
\]
where $P^*$ is defined using the transpose matrix of $A$. Consequently, $P$ has no root $(z_1,\dots,z_n)$ with $\lvert z_i\rvert >1$ for every $i$.

Proof of theorem 1. — Let $z$ be a root of $P$. Since the polynomial $P$ is good, so is the one-variable polynomial $F(T)=P(T,\dots,T)$. In particular, $F(z)=0$ implies $\lvert z\rvert \geq 1$. But the polynomial has a symmetry property, inherited by that of $P$, namely $ T^n F(1/T)=F^*(T)$, where $F^*$ is defined using the transpose matrix of $A$. Consequently, $F^*(1/z)=0$ and $\lvert 1/z\rvert \geq 1$. We thus have shown that $\lvert z\rvert=1$.




Wednesday, May 2, 2018

Combinatorics and probability : Greene, Nijenhuis and Wilf's proof of the hook length formula

I've never been very good at remembering representation theory, past the general facts that hold for all finite groups, especially the representation theory of symmetric groups.  So here are personal notes to help me understand the 1979 paper by Greene, Nijenhuis and Wilf, where they give a probabilistic proof of the hook length formula. If you already know what it is about, then you'd be quicker by browsing at the Wikipedia page that I just linked to!

Heroes of this story are partitions, Ferrers diagrams, and Young tableaux. Let us first give definitions.

A partition of an integer $n$ is a decreasing (US: non-increasing) sequence of integers $\lambda =(\lambda_1\geq \lambda_2\geq \dots\geq \lambda_m)$ of strictly positive integers such that $|\lambda|=\lambda_1+\dots+\lambda_m=n$.

The Ferrrers diagram $F(\lambda)$ of this partition $\lambda$ is the stairs-like graphic representation consisting of the first quadrant cells indexed by integers $(i,j)$, where $1\leq i\leq m$ and $1\leq j\leq \lambda_i$. Visualizing a partition by means of its Ferrers diagram makes clear that there is an involution on partitions, $\lambda\mapsto \lambda^*$, which, geometrically consists in applying the symmetry with respect to the diagonal that cuts the first quadrant. In formulas, $j\leq \lambda_i$ if and only if $i\leq \lambda_j^*$.

A Young tableau of shape $\lambda$ is an enumeration of the $n$ cells of this Ferrers diagram such that the enumeration strictly increases in rows and columns.

There is a graphic way of representing Ferrers diagrams and Young tableaux — the tradition says it's the Soviet way — consisting in rotating the picture by 45° ($\pi/4$) to the left and viewing the first quadrant as a kind of bowl in which $n$ balls with diameter $1$ are thrown from the top.

The hook length formula is a formula for the number $f_\lambda$ of Young tableaux of given shape $\lambda$.

For every $(i,j)$ such that $1\leq i\leq m$ and $1\leq j\leq \lambda_i$, define its hook $H_{ij}$ to be the set of cells in the diagram that are either above it, or on its the right — in formula, the set of all pairs $(a,b)$ such that $a=i$ and $j\leq b\leq \lambda_i$, or $i\leq a\leq m$ and $b=j\leq \lambda_a$. Let $h_{ij}$ be the cardinality of the hook $H_{ij}$.

Lemma. — One has $h_{a,b} = (\lambda_a-b)+(\lambda_b^*-a) +1$.

Indeed, $\lambda_a-b$ is the number of cells above $(a,b)$ in the Ferrers diagram of $\lambda$, excluding $(a,b)$, while $(\lambda_b^*-a)$ is the number of cells on the right of $(a,b)$.

Theorem (Frame, Thrall, Robinson; 1954). — Let $n$ be an integer and let $\lambda$ be a partition of $n$. The number of Young tableaux of shape $\lambda$ is given by 
\[ f_\lambda = \frac{n!}{\prod_{(i,j)\in F(\lambda)} h_{ij}}. \]

From the point of view of representation theory, partitions of $n$ are in bijection with conjugacy classes of elements in the symmetric group $\mathfrak S_n$ (the lengths of the orbits of a permutation of $\{1,\dots,n\}$ can be sorted into a partition of $n$, and this partition characterizes the conjugacy class of the given permutation). Then, to each partition of $n$ corresponds an irreducible representation of $\mathfrak S_n$, and $f_\lambda$ appears to be its dimension. (In a future post, I plan to explain this part of the story.)

As already said, the rest of this post is devoted to explaining the probabilistic proof due to Greene, Nijenhuis, and Wilf. (Aside: Nijenhuis is a Dutch name that should pronounced roughly like Nay-en uys.)

A corner of a Ferrers diagram is a cell $(i,j)$ which is both on top of its column, and on the right of its row; in other words, it is a cell whose associated hook is made of itself only. A bit of thought convinces that a corner can be removed, and furnishes a Ferrers diagram with one cell less. Conversely, starting from a Ferrers diagram with $n-1$ cells, one may add a cell on the boundary so as to get a Ferrers diagram with $n$ cells. In the partition point of view, either one part gets one more item, or there is one more part, with only one item. In a Young tableau with $n$ cells, the cell numbered $n$ is at a corner, and removing it furnishes a Young tableau with $n-1$ cells; conversely, starting from a Young tableau with $n-1$ cells, one can add a cell so that it becomes a corner of the new tableau, and label it with $n$.

Let $P(\lambda_1,\dots,\lambda_m)$ be the number on the right hand side of the Frame-Thrall-Robinson formula. By convention, it is set to be $0$ if $(\lambda_1,\dots,\lambda_m)$ does not satisfy $\lambda_1\geq\dots\geq\lambda_m\geq 1$. By induction, one wants to prove
\[ P(\lambda_1,\dots,\lambda_m) = \sum_{i=1}^m P(\lambda_1,\dots,\lambda_{i-1},\lambda_i-1,\lambda_{i+1},\dots,\lambda_m). \]
For every partition $\lambda$, set
\[ p_i(\lambda_1,\dots,\lambda_m) = \frac{P(\lambda_1,\dots,\lambda_{i-1},\lambda_i-1,\lambda_{i+1},\dots,\lambda_m)}{P(\lambda_1,\dots,\lambda_m)}. \]
We thus need to prove
\[ \sum_{i=1}^m p_i(\lambda)=1; \]
which we will do by interpreting the $p_i(\lambda)$ as the probabilities of disjoint events.

Given the Ferrers diagram $F(\lambda)$, let us pick, at random, one cell $(i,j)$, each of them given equal probability $1/n$; then we pick a new cell, at random, in the hook of $(i,j)$, each of them given equal probability $1/(h_{ij}-1)$, etc., until we reach a corner of the given diagram. Such a trial defines a path in the Ferrers diagram, ending at a corner $(a,b)$; its projections  are denoted by $A=\{a_1<a_2<\dots\}$ and $B=\{b_1<b_2<\dots\}$. Let $p(a,b)$ be the probability that we reach the corner $(a,b)$; let $q(A,B)$ be the probability that its projections be $A$ and $B$ conditioned to the hypothesis that it start at $(\inf(A),\inf(B))$.

Lemma. — Let $A,B$ be sets of integers, let $a=\sup(A)$ and let $b=\sup(B)$;  assume that $(a,b)$ is a corner of $\lambda$. One has \[ q(A,B)= \prod_{\substack{i\in A\\ i\neq a}} \frac1{h_{i,b}-1} \prod_{\substack{j\in B\\ j\neq b}} \frac1{h_{a,j}-1}.\]

We argue by induction on the cardinalities of $A$ and $B$. If $A=\{a\}$ and $B=\{b\}$, then $q(A,B)=1$, since both products are empty; this proves the formula in this case. As above, let $a_1<a_2<\dots$ be the enumeration of the elements of $A$ and $b_1<b_2<\dots$ be that for $B$; let also $A'=A\setminus\{a_1\}$ and $B'=B\setminus\{b_1\}$. By construction of the process, after having chosen the initial cell $(a_1,b_1)$,  it either goes on above the initially chosen cell $(a_1,b_1)$, hence at $(a_1,b_2)$, or on its right, that is, at $(a_2,b_1)$. One thus has
\[ \begin{align*}
q(A,B) =   \mathbf P(A,B\mid a_1,b_1)
& =  \mathbf P(a_1,b_1,b_2\mid a_1,b_1) \mathbf P(A,B\mid a_1,b_1,b_2)
+ \mathbf P(a_1,a_2,b_1\mid a_1,b_1) \mathbf P(A,B\mid a_1,a_2,b_1)\\
&= \frac1{f_{a_1,b_1}-1} \mathbf P(A,B'\mid a_1,b_2)
+ \frac1{f_{a_1,b_1}-1} \mathbf P(A',B\mid a_2,b_1) \\
&= \frac1{h_{a_1,b_1}-1}\left( q(A',B) + q(A,B') \right). \end{align*}
\]
By induction, we may assume that the given formula holds for $(A',B)$ and $(A,B')$. Then, one has
\[ \begin{align*}
q(A,B) & = \frac1{h_{a_1,b_1}-1} \left(
 \prod_{\substack{i\in A'\\ i\neq a}} \frac1{h_{i,b}-1} \prod_{\substack{j\in B \\ j\neq b}} \frac1{h_{a,j}-1}
+
 \prod_{\substack{i\in A\\ i\neq a}} \frac1{h_{i,b}-1} \prod_{\substack{j\in B' \\ j\neq b}} \frac1{h_{a,j}-1}\right) \\
& =\frac{  (h_{a_1,b}-1)+(h_{a,b_1}-1)}{h_{a_1,b_1}-1}
 \prod_{\substack{i\in A\\ i\neq a}} \frac1{h_{i,b}-1} \prod_{\substack{j\in B \\ j\neq b}} \frac1{h_{a,j}-1},
\end{align*}
\]
which implies the desired formula once one remembers that
\[ h_{a,b_1} + h_{a_1,b}
= h_{a_1,b_1} + h_{a,b}=  h_{a_1,b_1}+1\]
since $(a,b)$ is a corner of $F(\lambda)$.

Proposition. — Let $(a,b)$ be a corner of the diagram $F(\lambda)$; one has $p(a,b)=p_a(\lambda)$. (Note that $b=\lambda_a$.)

Write $F_a(\lambda)$ for the Ferrers diagram with corner $(a,b)$ removed. Its $(i,j)$-hook is the same as that of $F(\lambda)$ if $i\neq a$ and $j\neq b$; otherwise, it has one element less. Consequently, writing $h'_{i,j}$ for the cardinalities of its hooks, one has
\[\begin{align*}
p_a(\lambda) &= \frac1n \frac{\prod_{(i,j)\in F(\lambda)}h_{i,j}}{\prod_{(i,j)\in F_a(\lambda)} h'_{i,j}} \\ &= n \prod_{i<a} \frac{h_{i,b}}{h_{i,b}-1} \prod_{j<b}\frac{h_{a,j}}{h_{a,j}-1}\\
&=\frac1n \prod_{i<a}\left(1+ \frac1{h_{i,b}-1}\right) \prod_{j<b} \left(1+\frac1{h_{a,j}-1}\right). \end{align*}
\]
Let us now expand the products. We get
\[
p_a(\lambda) = \frac1n \sum_{\sup(A)<a} \sum_{\sup(B)<b} \prod_{i\in A} \frac1{h_{i,b}-1}\prod_{j\in B}\frac1{h_{a,j}-1},
\]
where $A$ and $B$ range over the (possibly empty) subsets of $\{1,\dots,n\}$ satisfying
the given conditions $\sup(A)<a$ and $\sup(B)<b$. (Recall that, by convention, or by definition, one has $\sup(\emptyset)=-\infty$.) Consequently, one has
\[ \begin{align*}
p_a(\lambda) & =\frac1n \sum_{\sup(A)=a} \sup_{\sup(B)=b} \prod_{\substack{i\in A\\ i\neq a}}
 \frac1{h_{i,b}-1} \prod_{\substack{j\in B \\ j\neq b}} \frac1{h_{a,j}-1} \\
& = \frac1n \sum_{\sup(A)=a} \sum_{\sup(B)=b} q(A,B) \\
& = \sum_{\sup(A)=a} \sum_{\sup(B)=b} \mathbf P (A,B ) \\
& = p(a,b),
\end{align*}
\]
as claimed.

Now, every trial has to end at some corner $(a,b)$, so that
\[ \sum_{\text{$(a,b)$ is a corner}} p(a,b) = 1.\]
On the other hand, if $(a,b)$ is a corner, then $b=\lambda_a$, while if $(a,\lambda_a)$ is not a corner, then $P_a(\lambda)=0$. We thus get $\sum_a P_a(\lambda)=P(\lambda)$, as was to be shown.

Wednesday, February 7, 2018

Contemporary homological algebra — Ignoramus et ignorabimus (?)

The title of this post is a quotation of Emil Dubois-Reymond (1818-1896), a 19th century German physiologist, and the elder brother of the mathematician Paul Dubois-Reymond. Meaning we are ignorant, and we will remain ignorant, it adopts a pessimistic point of view on science, which would have intrinsic limitations. As such, this slogan has been quite opposed by David Hilbert who declared, in 1900, at the International congress of mathematicians, that there is no ignorabimus in mathematics. (In fact, there is some ignorabimus, because of Gödel's incompleteness theorem, but that is not the subject of this post.)

I would like to discuss here, in a particularly informal way, some frustration of myself relative to homological algebra, in particular to its most recent developments. I am certainly ill-informed on those matters, and one of my goals is to clarify my own ideas, my expectations, my hopes,...

This mere existence of this post is due to the kind invitation of a colleague of the computer science department working in (higher) category theory, namely François Metayer, who was interested to understand my motivation for willing to understand this topic.


Let me begin with a brief historical summary of the development of homological algebra, partly borrowed from Charles Weibel's History of homological algebra.
  • B. Riemann (1857), E. Betti (1871), H. Poincaré (1895) define homology numbers. 
  • E. Noether (1925) introduces abelian groups, whose elementary divisors, recover the previously defined homology numbers.
  • J. Leray (1946) introduces sheaves, their cohomology, the spectral sequence... 
  • During the years 1940–1955, under the hands of Cartan, Serre, Borel, etc., the theory develops itself in various directions (cohomology of groups, new spectral sequences, etc.).
  • In their foundational book, Homological algebra, H. Cartan and S. Eilenberg (1956) introduce derived functors, projective/injective resolutions,...
  • Around 1950, A. Dold, D. Kan, J. Moore, D. Puppe introduce simplicial methods. D. Kan introduces adjoint functors.
  • A. Grothendieck, in Sur quelques points d'algèbre homologique (1957), introduces general abelian categories, as well as convenient axioms that guarantee the existence of enough injective objects, thus giving birth to a generalized homological algebra.
  • P. Gabriel and M. Zisman (1967) developed the abstract calculus of fractions in categories, and proved that the homotopy category of topological spaces coincides with that of simplicial sets.
  • J.-L. Verdier (1963) defines derived categories. This acknowledges that objects give rise to, say, injective resolutions which are canonical up to homotopy, and that the corresponding complex is an object in its own right, that has to be seen as equivalent to the initial object.  The framework is that of triangulated categories. Progressively, derived categories came to play an important rôle in algebraic geometry (Grothendieck duality, Verdier duality, deformation theory, intersection cohomology and perverse sheaves, the Riemann–Hilbert correspondence, mirror symmetry,...) and representation theory.
  • D. Quillen (1967) introduces model categories, who allow a parallel treatment of homological algebra in linear contexts (modules, sheaves of modules...) and non-linear ones (algebraic topology)... This is completed by A. Grothendieck's (1991) notion of derivators.
  • At some point, the theory of dg-categories appears, but I can't locate it precisely, nor do I understand precisely its relation with other approaches.
  • A. Joyal (2002) begins the study of quasi-categories (which were previously defined by J. M. Boardman and R. M. Vogt, 1973). Under the name of $(\infty,1)$-categories or $\infty$-categories, these quasi-categories are used extensively in Lurie's work (his books Higher topos theory, 2006; Higher algebra, 2017; the 10+ papers on derived algebraic geometry,...).
My main object of interest (up to now) is “classical” algebraic geometry, with homological algebra as an important tool via the cohomology of sheaves, and while I have barely used anything more abstract that cohomology sheaves (almost never complexes), I do agree that there are three main options to homological algebra: derived categories, model categories, and $\infty$-categories.

While I am not absolutely ignorant of the first one (I even lectured on them), the two other approaches still look esoteric to me and I can't say I master them (yet?). Moreover, their learning curve seem to be quite steep (Lurie's books totalize more than 2000 pages, plus the innumerable papers on derived algebraic geometry, etc.) and I do not really see how an average geometer should/could embark in this journey.

However, I believe that this is now a necessary journey, and I would like to mention some recent theorems that support this idea.

First of all, and despite its usefulness, the theory of triangulated/derived categories has many defects. Here are some of them:
  • There is no (and there cannot be any) functorial construction of a cone; 
  • When a triangulated category is endowed with a truncation structure, there is no natural functor from the derived category of its heart to the initial triangulated category; 
  • Derived categories are not well suited for non-abelian categories (filtered derived categories seem to require additional, non-trivial, work, for example);
  • Unbounded derived functors are often hard to define: we now dispose of homotopically injective resolutions (Spaltenstein, Serpé, Alonso-Tarrió et al.), but unbounded Verdier duality still requires some unnatural hypotheses on the morphism, for example.
Three results, now.

The first theorem I want to mention is due to M. Greenberg (1966). Given a scheme $X$ of finite type over a  complete discrete valuation ring $R$ with uniformizer $\pi$, there exists an integer $a\geq 1$, such that for any integer $n\geq1$, a point $x\in X(R/\pi^n)$ lifts to $X(R)$ if and only if it lifts to $X(R/\pi^{an})$.

It may be worth stating it in more concrete terms. Two particular cases of such a ring $R$ are the ring $R[[t]]$ of power series over some field $k$, then $\pi=t$, and the ring $\mathbf Z_p$ of $p$-adic numbers (for some fixed prime number $p$), in which case one has $\pi=p$. It is then important to consider the case of affine scheme. Then $X=V(f_1,\dots,f_m)$ is defined by the vanishing of a finite family $f_1,\dots,f_m$ of polynomials in $R[T_1,\dots,T_n]$ in $n$ variables, so that, for any ring $A$, $X(A)$ is the set of solutions in $A^n$ of the system $f(T_1,\dots,T_n)=\dots=f_m(T_1,\dots,T_n)=0$.  By reduction modulo $\pi^r$, a solution in $R^r$ gives rise to a solution in $R/\pi^r$, and Greenberg's result is about the converse: given a solution $x$ in $R/\pi^r$, how do decide whether it is a reduction of a solution in $R$. A necessary condition is that $x$ lifts to a solution in $R/\pi^s$, for every $s\geq r$. Greenberg's theorem asserts that it is sufficient that $x$ lift to a solution in $R/\pi^{ar}$, for some integer $a\geq 1$ which does not depend on $X$.

The proof of this theorem is non-trivial, but relatively elementary. After some preparation, it boils down to Hensel's lemma or, equivalently, Newton's method for solving equations.
However, it seems to me that there should be an extremely conceptual way to prove this theorem, based on general deformation theory such as the one developed by Illusie (1971). Namely, obstructions to lifting $x$ are encoded by various cohomology classes, and knowing that it lifts enough should be enough to see — on the nose — that these obstructions vanish.

The second one is about cohomology of Artin stacks. Y. Laszlo and M. Olsson (2006) established the 6-operations package for $\ell$-adic sheaves on Artin stacks, but their statements have some hypotheses which look a bit unnatural. For example, the base scheme $S$ needs to be such that all schemes of finite type have finite $\ell$-cohomological dimension — this forbids $S=\operatorname{Spec}(\mathbf R)$. More recently, Y. Liu and W. Zheng developed a more general theory, apparently devoid of restrictive hypotheses, and their work builds on $\infty$-categories, more precisely, a stable $\infty$-category enhancing the unbounded derived category. On page 7 of their paper, they carefully explain why derived categories are unsufficient to take care of the necessary descent datas, but I can't say I understand their explanation yet...

The last one is about the general formalism of 6-operations. While it is clear what these 6 operations should reflect (direct and inverse images; proper direct images and extraordinary inverse images; tensor product, internal hom), the list of the properties they should satisfy is not clear at all (to me). In the case of coherent sheaves, there is such a formulaire, written by A. Grothendieck itself on the occasion of a talk in 1983, but it is quite informal, and not at all a general formalism. Recently, F. Hörmann proposed such a formalism  (2015–2017), based on Grothendieck's theory of derivators.

Now, how should the average mathematician embark in learning these theories?

Who will write the analogue of Godement's book for the homological algebra of the 21st century? Can we hope that it be shorter than 3000 pages?

I hope to find, some day, some answer to these questions, and that they will allow to hear with satisfaction the words of Hilbert: Wir müssen wissen, wir werden wissen.

Wednesday, March 22, 2017

Warning! — Theorems ahead

Don't worry, no danger ahead! — This is just a short post about the German mathematician Ewald Warning and the theorems that bear his name.

It seems that Ewald Warning's name will be forever linked with that of Chevalley, for the Chevalley-Warning theorem is one of the rare modern results that can be taught to undergraduate students; in France, it is especially famous at the Agrégation level. (Warning published a second paper, in 1959, about the axioms of plane geometry.)

Warning's paper, Bemerkung zur vorstehenden Arbeit von Herrn Chevalley (About a previous work of Mr Chevalley), has been published in 1935 in the Publications of the mathematical seminar of Hamburg University (Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg), just after the mentioned paper of Chevalley. Emil Artin had a position in Hamburg at that time, which probably made the seminar very attractive; as a matter of fact, the same 1935 volume features a paper of Weil about Riemann-Roch, one of Burau about braids, one of Élie Cartan about homogeneous spaces, one of Santalo on geometric measure theory, etc.

1. The classic statement of the Chevalley—Warning theorem is the following.

Theorem 1. – Let $p$ be a prime number, let $q$ be a power of $p$ and let $F$ be a field with $q$ elements. Let $f_1,\dots,f_m$ be polynomials in $n$ variables and coefficients in $F$, of degrees $d_1,\dots,d_m$; let $d=d_1+\dots+d_m$. Let $Z=Z(f_1,\dots,f_m)$ be their zero-set in $F^n$. If $d<n$, then $p$ divides $\mathop{\rm Card}(Z)$.

This is really a theorem of Warning, and Chevalley's theorem was the weaker consequence that if $Z\neq\emptyset$, then $Z$ contains at least two points. (In fact, Chevalley only considers the case $q=p$, but his proof extends readily.) The motivation of Chevalley lied in the possibility to apply this remark to the reduced norm of a possibly noncommutative finite field (a polynomial of degree $d$ in $d^2$ variables which vanishes exactly at the origin), thus providing a proof of Wedderburn's theorem.

a) Chevalley's proof begins with a remark. For any polynomial $f\in F[T_1,\ldots,T_n]$, let $f^*$ be the polynomial obtained by replacing iteratively $X_i^q$ by $X_i$ in $f$, until the degree of $f$ in each variable is $<q$. For all $a\in F^n$, one has $f(a)=f^*(a)$; moreover, using the fact that a polynomial in one variable of degree $<q$ has at most $q$ roots, one proves that if $f(a)=0$ for all $a\in F^n$, then $f^*=0$.

Assume now that $Z$ contains exactly one point, say $a\in F^n$, let $f=\prod (1-f_j^{q-1})$, let $g_a=\prod (1-(x_i-a_i)^{q-1})$. Both polynomials take the value $1$ at $x=a$, and $0$ elsewhere; moreover, $g_a$ is reduced. Consequently, $f^*=g$. Then
$$ (q-1)n=\deg(g_a)=\deg(f^*)\leq \deg(f)=(q-1)\sum \deg(f_j)=(q-1)d, $$
contradicting the hypothesis that $d<n$.

b) Warning's proof is genuinely different. He first defines, for any subset $A$ of $F^n$ a reduced polynomial $g_A=\sum_{a\in A} g_a=\sum_{a\in A}\prod (1-(x_i-a_i)^{q-1})$, and observes that $g_A(a)=1$ if $a\in A$, and $g_A(a)=0$ otherwise.
Take $A=Z$, so that $f^*=g_Z$. Using that $\deg(f^*)\leq \deg(f)=(q-1)d$ and the expansion
$ (x-a)^{q-1} = \sum_{i=0}^{q-1} x^i a^{q-1-i}$, Warning derives from the equality $f^*=g_Z$
the relations
\[ \sum_{a\in Z} a_1^{\nu_1}\dots a_n^{\nu_n}=0, \]
for all $(\nu_1,\dots,\nu_n)$ such that $0\leq \nu_i\leq q-1$ and $\sum\nu_i <(q-1)(n-d)$. The particular case $\nu=0$ implies that $p$ divides $\mathop{\rm Card}(Z)$. More generally:

Proposition 2. — For every polynomial $\phi\in F[T_1,\dots,T_n]$ of reduced degree $<(q-1)(n-d)$, one has $\sum_{a\in A} \phi(a) = 0$.

c) The classic proof of that result is even easier. Let us recall it swiftly. First of all, for every integer $\nu$ such that $0\leq \nu <q$, one has $\sum_{a\in F} a^\nu=0$. This can be proved in many ways, for example by using the fact that the multiplicative group of $F$ is cyclic; on the other hand, for every nonzero element $t$ of $F$, the change of variables $a=tb$ leaves this sum both unchanged and multiplied by $t^\nu,$ so that taking $t$ such that $t^\nu\neq 1$, one sees that this sum vanishes. It follows from that that for every polynomial $f\in F[T_1,\dots,T_n]$ whose degree in some variable is $\lt;q-1$, one has $\sum_{a\in F^n} f(a)=0$. This holds in particular if the total degree of $f$ is $\lt; (q-1)n$.
Taking $f$ as above proves theorem 1.

2. On the other hand there is a second Warning theorem, which seems to be absolutely neglected in France. It says the following:

Theorem 3. — Keep the same notation as in theorem 1. If $Z$ is nonempty, then $\mathop{\rm Card}(Z)\geq q^{n-d}$.

To prove this result, Warning starts from the following proposition:

Proposition 4. – Let $L,L'$ be two parallel subspaces of dimension $d$ in $F^n$. Then $\mathop{\rm Card}(Z\cap L)$ and $\mathop{\rm Card}(Z\cap L')$ are congruent modulo $p$.

Let $r=n-d$. Up to a change of coordinates, one may assume that $L=\{x_1=\dots=x_{r}=0\}$ and $L'=\{x_1-1=x_2=\dots=x_{r}=0\}$. Let
\[ \phi = \frac{1-x_1^{q-1}}{1-x_1} (1-x_2^{q-1})\cdots (1-x_{r}^{q-1}).\]
This is a polynomial of total degree is $(q-1)r-1<(q-1)(n-d)$. For $a\in F^n$, one has $\phi(a)=1$ if $a\in L$, $\phi(a)=-1$ if $a\in L'$, and $\phi(a)=0$ otherwise. Proposition 4 thus follows from proposition 2. It is now very easy to prove theorem 3 in the particular case where there exists one subspace $L$ of dimension $d$ such that $\mathop{\rm Card}(Z\cap L)\not\equiv 0\pmod p$. Indeed, by proposition 4, the same congruence will hold for every translate $L'$ of $L$. In particular, $\mathop{\rm Card}(Z\cap L')\neq0$ for every translate $L'$ of $L$, and there are $q^{n-d}$ distinct translates.

To prove the general case, let us choose a subspace $M$ of $F^n$ of dimension $s\leq d$ such that
$\mathop{\rm Card}(Z\cap M)\not\equiv 0\pmod p$, and let us assume that $s$ is maximal.
Assume that $s< d$. Let $t\in\{1,\dots,p-1\}$ be the integer such that $\mathop{\rm Card}(Z\cap M)\equiv t\pmod p$. For every $(s+1)$-dimensional subspace $L$ of $F^n$ that contains $M$, one has $\mathop{\rm Card}(Z\cap L)\equiv 0\pmod p$, by maximaility of $s$, so that $Z\cap (L\setminus M)$ contains at least $p-t$ points. Since these subspaces $L$ are in 1-1 correspondence with the lines of the quotient space $F^n/M$, their number is equal to $(q^{n-s}-1)/(q-1)$. Consequently,
\[ \mathop{\rm Card}(Z) = \mathop{\rm Card}(Z\cap M) + \sum_L \mathop{\rm Card}(Z\cap (L\setminus M))
\geq t + (p-t) \frac{q^{n-s}-1}{q-1} \geq q^{n-s-1}\geq q^{n-d}, \]
as was to be shown.

3. Classic theorems seem to an everlasting source of food for thought.

a) In 1999, Alon observed that Chevalley's theorem follows from the Combinatorial Nullstellensatz he had just proved. On the other hand, this approach allowed Brink (2011) to prove a similar result in general fields $F$, but restricting the roots to belong to a product set $A_1\times\dots\times A_n$, where $A_1,\dots,A_n$ are finite subsets of $F$ of cardinality $q$. See that paper of Clarke, Forrow and Schmitt for further developments, in particular a version of Warning's second theorem.

b) In the case of hypersurfaces (with the notation of theorem 1, $m=1$), Ax proved in 1964 that the cardinality of $Z$ is divisible not only by $p$, but by $q$. This led to renewed interest in the following years, especially in the works of Katz, Esnault, Berthelot, and the well has not dried up yet.

c) In 2011, Heath-Brown published a paper where he uses Ax's result to strengthen the congruence modulo $p$ of proposition 4 to a congruence modulo $q$.

d) By a Weil restriction argument, a 1995 paper of Moreno-Moreno partially deduces the Chevalley-Warning theorem over a field of cardinality $q$ from its particular case over the prime field. I write partially because they obtain a divisibility by an expression of the form $p^{\lceil f \alpha\rceil}$, while one expects $q^{\lceil \alpha}=p^{f\lceil\alpha\rceil}$. However, the same argument allows them to obtain a stronger bound which does not involve not the degrees of the polynomials, but the $p$-weights of these degrees, that is the sum of their digits in their base $p$ expansions. Again, they obtain a divisibility by an expression of the form $p^{\lceil f\beta\rceil}$, and it is a natural question to wonder whether the divisibility by $p^{f\lceil\beta\rceil}$ can be proved.


Sunday, February 5, 2017

Counting points and counting curves on varieties — Tribute to Daniel Perrin

$\require{enclose}\def\VarC{\mathrm{Var}_{\mathbf C}}\def\KVarC{K_0\VarC}$
Daniel Perrin is a French algebraic geometer who turned 70 last year. He his also well known in France for his wonderful teaching habilities. He was one of the cornerstones of the former École normale supérieure de jeunes filles, before it merged in 1985 with the rue d'Ulm school. From this time remains a Cours d'algèbre which is a must for all the students (and their teachers) who prepare the agrégation, the highest recruitment process for French high schools. He actually taught me Galois theory (at École normale supérieure in 1990/1991) and Algebraic Geometry (the year after, at Orsay). His teaching restlessly stresses  the importance of examples. He has also been deeply involved in training future primary school teachers, as well as in devising the mathematical curriculum of high school students: he was responsible of the report on geometry. It has been a great honor for me to be invited to lecture during the celebration of his achievements that took place at Orsay on November, 23, 2016.

Diophantine equations are a source of numerous arithmetic problems. One of them has been put forward by Manin in the 80s and consists in studying the behavior of the number of solutions of such equations of given size, when the bound grows to infinity. A geometric analogue of this question considers the space of all curves with given degree which are drawn on a fixed complex projective, and is interested in their behavior when the degree tends to infinity. This was the topic of my lecture and is the subject of this post.

Let us first begin with an old problem, apparently studied by Dirichlet around 1840, and given a rigorous solution by Chebyshev and Cesáro around 1880: the probability that two integers be coprime is equal to $6/\pi^2$. Of course, there is no probability on the integers that has the properties one would expect, such as being invariant by translation, and the classical formalization of this problem states that the numbers of pairs $(a,b)$ of integers such that $1\leq a,b\leq n$ and $\gcd(a,b)=1$ grows as $n^2 \cdot 6/\pi^2$ when $n\to+\infty$,

This can be proved relatively easily, for example as follows. Without the coprimality condition, there are $n^2$ such integers. Now one needs to remove those pairs both of which entries are multiples of $2$, and there are $\lfloor n/2\rfloor^2$ of those, those where $a,b$ are both multiples of $3$ ($\lfloor n/3\rfloor^2$), and then comes $5$, because we have already removed those even pairs, etc. for all prime numbers. But in this process, we have removed twice the pairs of integers both of which entries are multiples of $2\cdot 3=6$, so we have to add them back, and then remove the pairs of integers both of which are multiples of $2\cdot 3\cdot 5$, etc. This leads to the following formula for
the cardinality $C(n)$ we are interested in:

$\displaystyle
 C(n) = n^2 - \lfloor\frac n2\rfloor^2 - \lfloor \frac n3\rfloor^2-\lfloor \frac n5\rfloor^2 - \dots
+ \lfloor \frac n{2\cdot 3}\rfloor^2+\lfloor\frac n{2\cdot 5}\rfloor^2+\dots
- \lfloor \frac n{2\cdot 3\cdot 5} \rfloor^2 - \dots $.

Approximating $\lfloor n/a\rfloor$ by $n/a$, this becomes

$\displaystyle
C(n) \approx  n^2 - \left(\frac n2\right)-^2 - \left (\frac n3\right)^2-\left( \frac n5\rfloor\right)^2 - \dots
+ \left (\frac n{2\cdot 3}\right)^2+\left(\frac n{2\cdot 5}\right)^2+\dots
- \left (\frac n{2\cdot 3\cdot 5} \right)^2 - \dots $

which we recognize as

$\displaystyle
C(n)\approx n^2 \left(1-\frac1{2^2}\right) \left(1-\frac1{3^2}\right)\left(1-\frac1{5^2}\right) \dots
=n^2/\zeta(2)$,

where $\zeta(2)$ is the value at $s=2$ of Riemann's zeta function $\zeta(s)$. Now, Euler had revealed the truly arithmetic nature of $\pi$ by proving in 1734 that $\zeta(2)=\pi^2/6$. The approximations we made in this calculation can be justified, and this furnishes a proof of the above claim.

We can put this question about integers in a broader perspective if we recall that the ring $\mathbf Z$ is a principal ideal domain (PID) and study the analogue of our problem in other PIDs, in particular for $\mathbf F[T]$, where $\mathbf F$ is a finite field; set $q=\operatorname{Card}(\mathbf F)$. The above proof can be adapted easily (with simplifications, in fact) and shows that number of pairs $(A,B)$ of monic polynomials of degrees $\leq n$ such that $\gcd(A,B)=1$ grows as $q^n(1-1/q)$ when $n\to+\infty$. The analogy becomes stronger if one observes that $1/(1-1/q)$ is the value at $s=2$ of $1/(1-q^{1-s})$, the Hasse-Weil zeta function of the affine line over $\mathbf F$.

What can we say about our initial question if we replace the ring $\mathbf Z$ with the PID $\mathbf C[T]$? Of course, there's no point in counting the set of pairs $(A,B)$ of coprime monic polynomials of degree $\leq n$ in $\mathbf C[T]$, because this set is infinite. Can we, however, describe this set? For simplicity, we will consider here the set $V_n$ of pairs of coprime monic polynomials of degree precisely $n$. If we identify a monic polynomial of degree $n$ with the sequence of its coefficients, we then view $V_n$ as a subset of $\mathbf C^{n}\times\mathbf C^n$. We first observe that $V_n$ is an Zariski open subset of $\mathbf C^{2n}$: its complement $W_n$ is defined by the vanishing of a polynomial in $2n$ variables — the resultant of $A$ and $B$.

When $n=0$, we have $V_0=\mathbf C^0=\{\mathrm{pt}\}$.

Let's look at $n=1$: the polynomials $A=T+a$ and $B=T+b$ are coprime if and only if $a\neq b$;
consequently, $V_1$ is the complement of the diagonal in $\mathbf C^2$.

For $n=2$, this becomes more complicated: the resultant of the polynomials $T^2+aT+b$ and $T^2+cT+d$ is equal to $a^2d-abc-adc+b^2-2bd+bc^2+d^2$; however, it looks hard to guess some relevant properties of $V_n$ (or of its complement) just by staring at this equation. In any case, we can say that $V_2$ is the complement in $\mathbf C^4$ of the union of two sets, corresponding of the degree of the gcd of $(A,B)$. When $\gcd(A,B)=2$, one has $A=B$; this gives the diagonal, a subset of $\mathbf C^4$ isomorphic to $\mathbf C^2$; the set of pairs of polynomials $(A,B)$ whose gcd has degree $1$ is essentially $\mathbf C\times V_1$: multiply a pair $(A_1,B_1)$ of coprime polynomials of degree $1$ by an arbitrary polynomial of the form $(T-d)$.
Consequently,
\begin{align}V_2&=\mathbf C^4 - \left( \mathbf C^2 \cup \mathbf C\times V_1\right)\\
&= \mathbf C^4 - \left( \enclose{updiagonalstrike}{\mathbf C^2}\cup \left(\mathbf C\times (\mathbf C^2-\enclose{updiagonalstrike}{\mathbf C})\right)\right)\\
&=\mathbf C^4-\mathbf C^3
\end{align}
if we cancel the two $\mathbf C^2$ that appear. Except that this makes no sense!

However, there is a way to make this computation both meaningful and rigorous, and it consists in working in the Grothendieck ring $\KVarC$ of complex algebraic varieties. Its additive group is generated by isomorphism classes of algebraic varieties, with relations of the form $[X]=[U]+[Z]$ for every Zariski closed subset $Z$ of an algebraic variety $X$, with complement $U=X-Z$. This group has a natural ring structure for which $[X][Y]=[X\times Y]$. Its unit element is the class of the point, $[\mathbf A^0]$ if one wishes. An important element of this ring $\KVarC$ is the class $\mathbf L=[\mathbf A^1]$ of the affine line. The natural map $e\colon \VarC\to \KVarC$ given by $e(X)=[X]$ is the universal Euler characteristic: it is the universal map from $\VarC$ to a ring such that $e(X)=e(X-Z)+e(Z)$ and $e(X\times Y)=e(X)e(Y)$, where $X,Y$ are complex varieties and $Z$ is a Zariski closed subset of $X$.

In particular, it generalizes the classical Euler characteristic, the alternate sum of the dimensions of the cohomology groups (with compact support, if one wishes) of a variety. A subtler invariant of $\KVarC$ is given by mixed Hodge theory: there exists a unique ring morphism $\chi_{\mathrm H}\KVarC\to\mathbf Z[u,v]$ such that for every complex variety $X$, $\chi_{\mathrm H}([X])$ is the Hodge-Deligne polynomial of $X$. In particular, if $X$ is projective and smooth, $\chi_{\mathrm H}([X])=\sup_{p,q} \dim h^q(X,\Omega^p_X) u^pv^q$. If one replaces the field of complex numbers with a finite field $\mathbf F$, one may actually count the numbers of $\mathbf F$-points of $X$, and this furnishes yet another generalized Euler characteristic.

The preceding calculation shows that $e(V_0)=1$, $e(V_1)=\mathbf L^2-\mathbf L$ and $e(V_2)=\mathbf L^4-\mathbf L^3$; more generally, one proves by induction that $e(V_n)=\mathbf L^{2n}-\mathbf L^{2n-1}$ for every integer $n\geq 0$.

Equivalently, one has $e(W_n)=\mathbf L^{2n-1}$ for all $n$. I have to admit that I see no obvious reason for the class of $W_n$ to be equal to that of an affine space. However, as Ofer Gabber and Jean-Louis Colliot-Thélène pointed out to me during the talk, this resultant is the difference of two homogeneous polynomials $p-q$ of degrees $d=2$ and $d+1=3$; consequently, the locus it defines is a rational variety — given $a,b,c$, there is generically a unique $t$ such that $p-q$ vanishes at $(at,bt,ct,t)$.

These three results have a common interpretation if one brings in the projective line $\mathbf P_1$. Indeed, pairs $(a,b)$ of coprime integers (up to $\pm1$) correspond to rational points on $\mathbf P_1$, and if $\mathbf F$ is a field, then pairs $(A,B)$ of coprime polynomials in $\mathbf F[T]$ correspond (up to $\mathbf F^\times$) to elements of $\mathbf P_1(\mathbf F(T))$.
In both examples, the numerical datum $\max(|a|,|b|)$ or $\max(\deg(A),\deg(B))$ is called the height of the corresponding point.

In the case of the ring $\mathbf Z$, or in the case of the ring $\mathbf F[T]$ where $\mathbf F$ is a finite field, one has an obvious but fundamental finiteness theorem: there are only finitely many points of $\mathbf P_1$ with bounded height. In the latter case, $\mathbf C[T]$, this naïve finiteness does not hold. Nevertheless, if one sees $\mathbf P_1(\mathbf C(T))$ as an infinite dimensional variety — one needs infinitely many complex numbers to describe a rational function, then the points of bounded height constitute what is called a bounded family, a “finite dimensional” constructible set.

The last two examples have a common geometric interpretation. Namely, $\mathbf F(T)$ is the field of functions of a projective smooth algebraic curve $C$ over $\mathbf F$; in fact, $C$ is the projective line again, but we may better ignore this coincidence. Then a point $x\in\mathbf P_1(\mathbf F(T))$
corresponds to a morphism $\varepsilon_x\colon C\to\mathbf P_1$, and the formula $H(x)=\deg(\epsilon_x^*\mathscr O(1))$ relates the height $H(x)$ of $x$ to the degree of the morphism $\varepsilon_x$.

Since the notion of height generalizes from $\mathbf P_1$ to projective spaces $\mathbf P_n$ of higher dimension (and from $\mathbf Q$ to general number fields), this suggests a general question. Let $V\subset\mathbf P_n$ be a projective variety over a base field $k$ hat can one say about the set of points $x\in V(k)$ such that $H(x)\leq B$, when the bound $B$ grows to $\infty$?
The base field $k$ can be either a number field, or the field of functions $\mathbf F(C)$ of a curve $C$ over a finite field $\mathbf F$, or the field of functions $\mathbf C(C)$ of a curve over the complex numbers. In the last two cases, the variety can even be taken to be constant, deduced from a variety $V_0$ over $\mathbf F$ or $\mathbf C$.

  1. When $k$ is a number field, this set is a finite set; how does its cardinality grows? This is a question that Batyrev and Manin have put forward at the end of the 80s, and which has attracted a lot of attention since.
  2. When $k=\mathbf F(C)$ is a function field over a finite field, this set is again a finite set; how does its cardinality grows? This question has been proposed by Emmanuel Peyre by analogy with the question of Batyrev and Manin.
  3. When $k=\mathbf C(C)$ is a function field over $\mathbf C$, this set identifies with a closed subscheme of the Grothendieck-Hilbert scheme of $V$; what can one say about its geometry, in particular about its class in $\KVarC$? Again, this question has been proposed by Emmanuel Peyre around 2000.

In a forthcoming post, I shall recall some results on these questions, especially the first one, and in particular explain an approach based on the Fourier summation formula. I will then explain a theorem proved with François Loeser where we make use of Hrushovski–Kazhdan's motivic Fourier summation formula in motivic integration to prove an instance of the third question.