Polynomial equations in function fields

TL;DR

This paper applies polynomial methods to bound the size of polynomial solution-free sets over finite fields, achieving exponential bounds that improve upon previous integer-based results.

Contribution

It introduces new bounds for solution-free polynomial sets over finite fields, extending polynomial method applications to new algebraic equations.

Findings

Bound of the form q^{cn} for some c<1 on solution-free polynomial sets

Contrast with integer case bounds that have only logarithmic savings

Applicable for equations with at least 2r^2+1 variables

Abstract

The breakthrough paper of Croot, Lev, Pach \cite{CLP} on progression-free sets in $\Z_4^n$ introduced a polynomial method that has generated a wealth of applications, such as Ellenberg and Gijswijt's solutions to the cap set problem \cite{EG}. Using this method, we bound the size of a set of polynomials over $\F_q$ of degree less than $n$ that is free of solutions to the equation $\sum_{i=1}^k a_if_i^r=0$, where the coefficients $a_i$ are polynomials that sum to 0 and the number of variables satisfies $k\geq 2r^2+1$. The bound we obtain is of the form $q^{cn}$ for some constant $c<1$. This is in contrast to the best bounds known for the corresponding problem in the integers, which offer only a logarithmic saving, but work already with as few as $k\geq r^2+1$ variables.

Equations

$$\sum _{i=1}^k{a}_i{f}_i^r=0$$

$$\overrightarrow{f^r}=(f_0^r,r{f}_0^{r-1}{f}_1,\dots ,f_{n-1}^r)\in {\mathbb{F}}_q^{(n-1)r+1}.$$

$$\Phi :(f_1,\dots ,f_k)\mapsto \sum _{i=1}^k{a}_i{f}_i^r$$

$$\overrightarrow{\Phi }:({\mathbb{F}}_q^n{)}^k\to {\mathbb{F}}_q^m$$

$$\overrightarrow{\Phi }(\overrightarrow{f_1},\dots ,\overrightarrow{f_k})=\overrightarrow{\Phi (f_1,\dots ,f_k)}.$$

$$\frac{mr}{kn}=\frac{(n-1)r^2+dr+r}{kn}\le \frac{r^2}{2r^2+1}+\frac{(d+1)r}{(2r^2+1)n}.$$

$$ϵ=ϵ(r)=\frac{1}{4(2r^2+1)}\in (0,1/2).$$

$$\left|A\right|\le kC(d,r,q)q^{c(ϵ,q)n}$$

$$\forall (f_1,\dots ,f_k)\in A^k,\Phi (f_1,\dots ,f_k)=0\iff f_1=\cdots =f_k$$

$$\forall (f_1,\dots ,f_k)\in A^k,\prod _{i=1}^m(1-{\Phi }_i^{q-1}(f_1,\dots ,f_k))=\sum _{f\in A}\prod _{j=1}^k{\delta }_f(f_j)$$

$$P(X_1,\dots ,X_k)=\sum _{j\in [k]}\sum _{p\in M_j}p(X_j)F_{j,p}(X_1,\dots ,X_{j-1},X_{j+1},\dots ,X_k).$$

$$p(f_1,\dots ,f_k)=\prod _{j\in [k]}{p}_j(f_{j,1},\dots ,f_{j,n})$$

$$P=\sum _{j=1}^k\sum _{p\in M_j}p(f_j)F_{j,p}(f_1,\dots ,f_{j-1},f_{j+1},\dots ,f_k).$$

$$\left|{\mathcal{M}}_{(q-1)m\ell /k,n}\right|\le \left|{\mathcal{M}}_{(q-1)n(1/2-ϵ),n}\right|\le q^n{e}^{-\frac{n{ϵ}^2}{2}}=q^{c(ϵ,q)n}$$

Full text

Let $R$ be a ring and $a_{1},\ldots,a_{k}$ be elements of $R$ which sum to 0, i.e. $\sum_{i=1}^{k}a_{i}=0$. Then the equation

[TABLE]

possesses a wealth of trivial solutions $(f_{1},\ldots,f_{k})$, namely constant tuples $(f,\ldots,f)$, even though it is not a translation-invariant equation. This suggests that if a subset $A\subset R$ is free of non-trivial solutions, then it should be small. For the ring $R=\mathbb{Z}$, this question was studied first by Smith [9], Keil [7] and Henriot [6]; they replaced the single equation by a system comprising the initial equation and a linear equation in order to ensure invariance under translation and dilation. Recently, Browning and Prendiville [1] showed that if $r=2$ and $k\geq 5$, and $A\subset[N]$ satisfies $\left\lvert A\right\rvert\gg N$ and $N$ is large enough, then equation (1) necessarily admits non-trivial solutions $(f_{1},\ldots,f_{k})$ in $A^{k}$. Their method relies on the transference principle. Further, Chow [2] proved that any relatively dense subset of the primes contains a solution to any equation of the form (1), as long as $k\geq r^{2}+1$.

Similarly, one may ask whether any dense subset $A$ of the ring $R=\mathbb{F}_{q}[t]$ is bound to contain a non-trivial solution to (1). In this note, we answer the question under a natural condition on the number of variables, namely $k\geq 2r^{2}+1$. In the function field setting, the polynomial method of Croot, Lev and Pach [3] can be fruitfully applied and delivers much stronger bounds than any method known in the integers. This was already noticed by Green [5] in the case of Sarközy’s theorem.

We now precisely state our main theorem. We fix a prime power $q$ and write $P_{q,n}$ for the set of polynomials of degree strictly less than $n$ over $\mathbb{F}_{q}$, so that $\left\lvert P_{q,n}\right\rvert=q^{n}$.

The aforementioned paper of Chow [2] implies that $k\geq r^{2}+1$ is sufficient in the integers, but the bound on the size of $A$ obtained by his analytic method is much weaker (we get a power saving, as opposed to his logarithmic saving).

We reduce the theorem to the following proposition, which is then tractable by the polynomial method of Croot-Lev-Pach.

We prove that Proposition 2 implies Theorem 1. Each polynomial $f=\sum_{i=0}^{n-1}f_{i}t^{i}\in P_{q,n}$ can be seen as a vector $\overrightarrow{f}=(f_{0},\ldots,f_{n-1})\in\mathbb{F}_{q}^{n}$. Now $f^{r}\in P_{q,(n-1)r+1}$ so we see it as the vector

[TABLE]

We notice that $\overrightarrow{f^{r}}=Q(\overrightarrow{f})$ where $Q$ is a polynomial map of degree $r$. Similarly, if $a\in\mathbb{F}_{q}[t]$ of degree at most $d$, we see that $f\mapsto\overrightarrow{af}\in\mathbb{F}_{q}^{n+d}$ is a polynomial map $\mathbb{F}_{q}^{n}\rightarrow\mathbb{F}_{q}^{n+d}$ (of degree 1). Thus,

[TABLE]

induces a polynomial map of degree $r$

[TABLE]

where $m=(n-1)r+d+1$ and

[TABLE]

We observe that if $A\subset P_{q,n}$ does not contain any non-trivial solution to (1), the set $\overrightarrow{A}=\{\overrightarrow{f}\mid f\in A\}\subset\mathbb{F}_{q}^{n}$ contains only trivial solutions $(\overrightarrow{f},\ldots,\overrightarrow{f})$ to the equation $\overrightarrow{\Phi}(\overrightarrow{f_{1}},\ldots,\overrightarrow{f_{k}})=0$.

Moreover, given that $k\geq 2r^{2}+1$, we have

[TABLE]

Hence if $n\geq 4(d+1)r$, we have $mr/k\leq(1/2-\epsilon)n$, with

[TABLE]

We can then apply Proposition 2 and obtain $\left\lvert A\right\rvert\leq q^{c(r,q)n}$ for some constant $c(r,q)=c^{\prime}(\epsilon(r),q)\in(0,1)$. Taking care separately of the small values of $n$, one can find a constant $C(d,r,q)\leq q^{4(d+1)r}$ such that the bound

[TABLE]

is valid for all $n$.

We now prove Proposition 2. We remark, in the spirit of Tao’s blog post [10], that the fact that

[TABLE]

implies that

[TABLE]

where $\delta_{f}(f_{j})$ is 0 if $f\neq f_{j}$ and 1 otherwise. We now recall the notion of slice-rank, as in Tao’s blog post or the article of Kleinberg, Sawin and Speyer [8]. Take a subset $A\subset\mathbb{F}_{q}^{n}$ and a map $P:A^{k}\rightarrow\mathbb{F}_{q}$. Let $\mathcal{M}$ be the set of functions $\mathbb{F}_{q}^{n}\rightarrow\mathbb{F}_{q}$. This set of functions is naturally in bijection with the set of polynomials in $\mathbb{F}_{q}[t_{1},\ldots,t_{n}]$ in which no indeterminate is raised to a power greater than $q-1$ (see for instance [5] for a proof of this bijection).

[TABLE]

The slice rank of $P$ is the minimum size $\sum_{j\in[k]}\left\lvert M_{j}\right\rvert$ of a polynomial cover.

We find that the slice-rank of the right-hand side of (2) is $\left\lvert A\right\rvert$; indeed, this is [10, Lemma 1]. The left-hand side of (2) is a polynomial $P$ in $k\times n$ variables denoted $f_{j,i}$ for $j\in[k]$ and $i\in[n]$, and its total degree is at most $(q-1)m\ell$. Now $P$ is a sum of monomials of the form

[TABLE]

where each $p_{j}$ is a monomial in $n$ variables. For each monomial $p$, by the pigeonhole principle, there exists $j\in[k]$ such that $\deg p_{j}\leq(q-1)m\ell/k$.

Being interested in $P$ as a function on $(\mathbb{F}_{q}^{n})^{k}$, we reduce it modulo the ideal $I$ generated by the polynomials $f_{j,i}^{q-1}-f_{j,i}$ for $i\in[n]$ and $j\in[k]$. We continue to use $P$ for the only polynomial in the class $P$ modulo $I$ which has degree at most $q-1$ in each variable $f_{j,i}$. Further, we denote by $\mathcal{M}_{d,n}$ the set of monomials in $n$ variables, of degree at most $q-1$ in each variable and at most $d$ in total.

We infer from the data above that there exist sets of monomials $M_{1},\ldots,M_{k}\subset\mathcal{M}_{(q-1)m\ell/k,n}$ and functions $F_{j,p}$ for $(j,p)\in[k]\times M_{j}$ such that

[TABLE]

Now $\left\lvert\mathcal{M}_{d,n}\right\rvert/q^{n}$ may be interpreted as the probability that the sum of $n$ independent, uniform random variables on $\{0,\ldots,q-1\}$ is at most $d$. To bound this probability, we use Hoeffding’s concentration inequality, which implies that

[TABLE]

where $c(\epsilon,q)=(1-\frac{\epsilon^{2}}{2\log q})\in(0,1)$. This implies that the slice-rank of $P$ is at most $kq^{c(\epsilon,q)n}$ and concludes the proof of Proposition 2.

In fact, we observe that the scope of our theorem encompasses more general equations than the diagonal equation (1), because Proposition 2 does not require any information on $\Phi$ other than its degree.