On the bias of cubic polynomials

TL;DR

This paper investigates the bias of cubic polynomials over finite fields, establishing a modified converse statement to a known decomposition result for degree 3 polynomials.

Contribution

It proves a modified version of the converse statement for the bias of cubic polynomials, extending previous results on polynomial decomposition over finite fields.

Findings

Bias of cubic polynomials can be characterized by their decompositions.

A modified converse statement holds for degree 3 polynomials.

The results connect polynomial bias with algebraic structure.

Abstract

Let $V$ be a vector space over a finite field $k=\mathbb{F} _q$ of dimension $n$. For a polynomial $P:V\to k$ we define the bias of $P$ to be $$b_1(P)=\frac {|\sum _{v\in V}\psi (P(V))|}{q^n}$$ where $\psi :k\to \mathbb{C} ^\star$ is a non-trivial additive character. A. Bhowmick and S. Lovett proved that for any $d\geq 1$ and $c>0$ there exists $r=r(d,c)$ such that any polynomial $P$ of degree $d$ with $b_1(P)\geq c$ can be written as a sum $P=\sum _{i=1}^rQ_iR_i$ where $Q_i,R_i:V\to k$ are non constant polynomials. We show the validity of a modified version of the converse statement for the case $d=3$.

Equations

$$b_1(P)=\frac{|{\sum }_{v\in V}\psi (P(V))|}{q^n}$$

$$P=\sum _{i=1}^r{l}_i{R}_i$$

$$a(P,l)=\sum _{v\in V(k)}\psi (P(v)+l(v))$$

$$\sum _{v\in P^{\perp }(k)}\psi (P(v+v^{\prime })+l(v+v^{\prime }))=0.$$

$$P(v)+l(v)=P(v+v_0)+C;v\in V.$$

$$P(v)=\sum _1^t{x}_i{x}_{i+t},v=(x_1,...,x_N)$$

$$P(v)=\sum _1^t{x}_i{x}_{i+t}+\alpha x_r^2,\alpha \in k^{\star },v=(x_1,...,x_N)$$

$$P=x_{1}R(x)$$

$$r_x=\sum _{y\in W(k)}\psi (R(x,y)).$$

$$a(P)=\sum _{y\in W(k)}1+\sum _{x\in k^{\star }}r(a)=q^{(N-1)}+\sum _{x\in k^{\star }}{r}_a$$

$$R(x,y)=\alpha x^2+xl(y)+Q(y)$$

$$|r_x|\le q^{N-1}{q}^{-r(Q)/2}\le q^{N-1-3/2}$$

$$P(\bar{x},y)=\sum _{i=1}^2{x}_i{R}^i(y)$$

$$R^i(\bar{x},y)=c^i(\bar{x})+l^i(\bar{x})(y)+Q^i(y),\bar{x}\in k^2,y\in W.$$

$$Q^{\bar{x}}=\sum _{i=1}^2{x}_i{Q}^i$$

$$r_{\bar{x}}=\sum _{y\in W(k)}\psi (P(\bar{x},y)).$$

$$a(P)=q^{N-2}+\sum _{\bar{a}\in k^2-(0)}{r}_{\bar{a}}$$

$$|r_{\bar{x}}|\le q^{N-6}$$

$$U=\{\bar{x}\in k^2-\{0\}|rank(Q^{\bar{x}})\le 12\}$$

$$P^{\prime }={\tilde{P}}^{\prime }\circ s^{\prime },s^{\prime }:V^{\prime }\to V^{\prime }/W^{\prime \prime }$$

$$U_1=\{\bar{x}\in k^2-\{0\}|rank(Q^{\bar{x}})\le 16+2u\}$$

$$P(\bar{x},y)=\sum _{i=1}^r{x}_i{R}^i(y)$$

$$R^i(\bar{x},y)=c^i(\bar{x})+l^i(\bar{x})(y)+Q^i(y),\bar{x}\in k^r,y\in W$$

$$Q^{\bar{x}}=\sum _{i=1}^r{x}_i{Q}^i$$

$$r_{\bar{x}}=\sum _{y\in W(k)}\psi (P(\bar{x},y)).$$

$$a(P)=q^{N-r}+\sum _{\bar{a}\in k^r-(0)}{r}_{\bar{a}}$$

$$U_0:=\{\bar{x}\in k^r-\{0\}|rank(Q^{\bar{x}})\le 2r+2+u_0\}$$

$$\mathcal{F}(F)={p_1}_{!}(p^{\star }({\mathcal{L}}_{AS}\otimes p_2^{\star }(F))$$

$$p(x,y)=xy,p_1(x,y)=x,p_2(x,y)=y,$$

$$\gamma :x\to x+\gamma ,\gamma \in k,x\in \mathbb{A}.$$

Full text

[TABLE]

where $\psi:k\to{\mathbb{C}}^{\star}$ is a non-trivial additive character.

A. Bhowmick and S. Lovett proved that for any $d\geq 1$ and $c>0$ there exists $r=r(d,c)$ such that any polynomial $P$ of degree $d$ with $b_{1}(P)\geq c$ can be written as a sum $P=\sum_{i=1}^{r}Q_{i}R_{i}$ where $Q_{i},R_{i}:V\to k$ are non constant polynomials. We show the validity of a modified version of the converse statement for the case $d=3$.

1. Introduction

Let $k=\mathbb{F}_{q},q=p^{m},k_{n}/k$ be the extension of degree $n$, $\bar{k}$ the algebraic closure of $k,tr_{k_{n}/k}:k_{n}\to k$ the trace. We fix a non-trivial additive character $\psi:k\to{\mathbb{C}}^{\star}$ and define $\psi_{n}:=\psi\circ tr_{k_{n}/k}:k_{n}\to{\mathbb{C}}^{\star}$.

Definition 1**.**

Let $V$ be a $k$-vector space and $PV\to k$ be a polynomial function depending on a finite number of variables. We define

$(a)$ $a_{n}(P)=\sum_{v\in V(k_{n})}\psi_{n}(P(v)),$

$(b)$ $\tilde{b}_{n}(P)=\frac{|a_{n}(P)|}{q^{ndim(V)}},$

$(c)$ $b_{n}(P)=-2\log_{q^{n}}(\tilde{b}_{n}(P))$ where $-log_{q^{n}}(0)=\infty$

and $(d)$ $b(P)=\liminf_{n}b_{n}(P)$.

We say that $b_{n}(P)$ is the bias of $P$ over $k_{n}$ and $b(P)$ is the bias of $P$.

Remark 1*.*

$a_{n}(V)$ is well defined since $a_{n}(P)=a_{n}(Q)$ if $P=f^{\star}(Q)$ where $f:V\to W$ is a linear map.

Definition 2**.**

For a non-zero homogeneous polynomial $P\in k[V^{\vee}]$ of degree $d\geq 2$ we define the algebraic rank $r(P)$ of $P$ as $2r$ where $r$ is the minimal number $r$ such that it is possible to write $P$ in the form

[TABLE]

where $l_{i},R_{i}\in\bar{k}[V^{\vee}]$ are homogeneous polynomials of positive degrees.

As shown in [BL] for any $\geq 1,c>0$ there exists $M(d,c)$ such that $r(P)\leq M(d,c)$ for any polynomial $P$ of degree $d$ with $b_{1}(P)\geq c$.

From now on we assume that $P$ is homogeneous of degree $d<p$.

Theorem 1**.**

$b(P)<\infty$.

Corollary 2**.**

For any $l\geq 1,d<p$ there exist $n(l,d)\in{\mathbb{Z}}_{+},c(l,d)>0$ such that for any polynomial $P:k^{l}\to k$ of degree $d$ we have $|a_{n}(P)|\geq c(l,d)$ for some $n$, $1\leq n\leq n(l,d)$.

The Corollary follows from the finiteness of $k$.

We want to express our gratitude to A. Braverman who helped with a proof of Lemma 10 and T. Schlank for a helpful discussions.

2. The quadratic case

In this section we assume that $q$ is odd. Let $V$ be a finite-dimensional $k$-vector space, $P:V\to k$ a quadratic polynomial,

$P^{\perp}=\{v\in V|Q(v+v^{\prime})=Q(v^{\prime})$ for all $v^{\prime}\in V$. Let $l:V\to k$ be a linear functional and

[TABLE]

Lemma 3**.**

$(a)$* If $l_{P^{\perp}}\neq 0$ then $a(P,l)=0$.*

$(b)$* If $l_{P^{\perp}}=0$ then $a(P,l)=0$ then $|r(P)-b(P)|\leq 1$.*

Proof. $(a)$. If $l_{P^{\perp}}\neq 0$ then for any $v^{\prime}\in V$ we have

[TABLE]

$(b)$. If $l_{P^{\perp}}=0$ then there exist $v_{0}\in V,C\in k$ such that

[TABLE]

So we can assume that $l=0$.

If $r=2t$ then there exists a system of coordinates $(x_{1},...,x_{N})$ on $V$ such that

[TABLE]

Since $\sum_{x,y\in k}=\psi(xy)=q$ we have $b_{1}(P)=r$.

If $r=2t+1$ then there exists a system of coordinates $(x_{1},...,x_{N})$ on $V$ such that

[TABLE]

In this case we have $b_{1}(P)=r-1/2$.

3. The cubic case.

Theorem 4**.**

For any $c>0$ there exist $t=t(c,q)>0$ such that $b(P)\leq t(c,q)$ for any cubic polynomial $P$ on a $k$-vector space $V$ with $r(P)\leq c$. Also there exists $N(c,q)$ such for any cubic polynomial $P$ on a $k$-vector space $V$ with $r(P)\leq c$ we have $b_{n}(P)\leq t(c,q)$ for some $n\leq N(c,q)$.

Conjecture 5**.**

One can choose $t(c)$ such that the statement of Theorem 4 is true for all finite fields of characteristic $>3$.

In this section we derive Theorem 4 from Corollary 2 and then prove Theorem 1 in the next section.

Proof.

We have to show that for any $T\geq 1$ there exists $N(T),n(T)$ and $b(T)>0$ such that for any $k$-vector space $V$ and a cubic polynomial $P:V\to k$ such that $r(P)\leq T$ there exists $n\leq n(T)$ such that $b_{n}(P)\geq b(T)$.

We fix $P$ of the form $P=\sum^{r}_{i=1}l_{i}R_{i}$ where $l_{i}$ are linear and $R_{i}$ quadratic polynomials. We can assume that the linear functions $l_{i}$ are linearly independent.

Let $W=\{v\in V|l_{i}(v)=0,1\leq i\leq r\}$ and $R^{\prime}_{i}$ be the restrictions of $R_{i}$ on $W$.

Consider first the case when $r=1$. We can assume that $V=k^{N},l_{1}(x_{1},...,x_{N})=x_{1}$. So

[TABLE]

where $R$ is a quadratic form and $W=k^{N-1}$. We write $v\in V$ in the form $v=(x,y),x\in k,y\in W(k)$.

For any $x\in k^{\star}$ we define

[TABLE]

By definition we have

[TABLE]

It is sufficient to prove the following result.

Lemma 6**.**

Either $|r_{x}|\leq q^{(N-3/2)}$ for all $x\in k^{\star}$ or there exist linear function $t_{1},t_{2},t_{3}$ on $V$ and a cubic form $Q$ on $k^{3}$ such that $P(v)=Q(t_{1}(v),t_{2}(v),t_{3}(v))$.

Proof. We write vectors of $v\in V$ in the form $v=(x,y),x\in k,y\in k^{N-1}$. Then

[TABLE]

where $\alpha\in k,l$ is a linear function and $Q$ is a quadratic form on $W$. Let $W^{\prime}\subset W$ be the kernel of $Q$.

We consider two cases.

If $r(Q)\geq 3$ then it follows from Lemma 3 that

[TABLE]

for all $x\in k^{\star}$ and therefore $b(P)=2$.

On the other hand if $r(Q)\leq 2$ then there is a subspace $W^{\prime}\subset W$ of codimension $2$ such that $Q(y+y^{\prime})=Q(y)$ for $y\in W,y^{\prime}\in W^{\prime}$. If $l_{W^{\prime}}\neq 0$ then $r_{x}=0$ forall $x\in k^{\star}$.

One the other hand if $l_{W^{\prime}}=0$ then $P=p^{\star}(\bar{P})$ where $\bar{P}$ is a polynomial of degree $3$ on $W/W^{\prime}$ and Theorem 4 follows from Corollary 2.

To simplify notations we consider in details only the case $r=2$. The general case is completely analogous.

We can assume that $V=k^{N}$ and $l_{i}(x_{1},...,x_{N})=x_{i},1\leq i\leq 2$. So $W\subset V$ is defined by the system of equation $\{x_{1}(v)=x_{2}(v)=0\}$. We denote elements of $W$ by $y$ and write $v\in V$ in the form $v=(\bar{x},y)$ where $\bar{x}=(x_{1},x_{2})$. Then we have

[TABLE]

where $R^{i}$ are polynomials of degree $2$. So there exist quadratic forms $c^{i}$ on $k^{2}$, linear maps $l^{i}\in Hom(k^{2},W^{\vee})$ and quadratic forms $Q^{i}$ on $W$ such that

[TABLE]

We define

[TABLE]

for $\bar{x}\in k^{2}-0$.

For any $\bar{x}\in k^{2}-0$ we define

[TABLE]

By definition we have

[TABLE]

If $rank(Q^{\bar{x}})\geq 12$ then (as follows from Lemma 3) we have

[TABLE]

Let

[TABLE]

and $R\subset k^{2}$ be the subspace generated by $U$. There are three possibilities for the dimension of $R$.

$(0)$. If $R=\{0\}$ then $rank(Q^{\bar{x}})\geq 12$ for all $\bar{x}\in k^{2}-\{0\}$ we have $a_{1}(P)\sim q^{N-2}$.

$(2)$. If $R=k^{2}$ choose linearly independent $u,u^{\prime}\in U$ and define $W^{\prime}={Q^{u}}^{\perp}\cap{Q^{u^{\prime}}}^{\perp}\subset W$. Then $W^{\prime}\subset V$ is a subset of codimention $\leq 26$ and all quadratic forms $Q^{\bar{x}},\bar{x}\in k^{2}$ are invariant under shifts by $w^{\prime}\in W^{\prime}$. Let $W^{\prime\prime}=ker(l^{1})\cap ker(l^{2})\cap W^{\prime}$. Then there exists a polynomial $\tilde{P}$ on $W/W^{\prime\prime}$ such that $P=\tilde{P}\circ s$ where $s:W\to W/W^{\prime\prime}$ is the projection. It follows from Corollary 2 that in this case the conclusion of Theorem 4 is true.

$(1)$. Now consider the case when $dim_{k}(R)=1$. We can then assume that $U=\{(0,x)\},x\in k$. Let $V^{\prime}=\{(x_{1},...,x_{n})|x_{1}=0\}$ and $P^{\prime}$ be the restriction of $P$ on $V^{\prime}$. It is clear that there exists a cubic polynomial $\tilde{P}^{\prime}$ on $V^{\prime}/W^{\prime\prime}$ such that

[TABLE]

Since $dim(V^{\prime}/W^{\prime\prime})\leq 26$ it follows from Corollary 2 there exist $n\leq n(26,3)$ such that $|a_{n}(P^{\prime})|\geq c$, $c=c(26,3)$.

Let $u=-loq_{q}(c)$ and

[TABLE]

and $R^{\prime}\subset k^{2}$ be the subspace generated by $U^{\prime}$.

If $R_{1}=k^{2}$ then we finish the proof of Theorem 2 as in the case $(2)$. On the other hand if $dim(R_{1})=R$ we see that $|a_{n}(P)-a_{n}(P^{\prime})|\leq q^{(N-u-1)n}$ and therefore as before it follows from the definition of $u$ that in this case the conclusion of Theorem 4 is also true.

We only indicate a way to start the proof of the result for the general $r$ and leave details to the reader.

We can assume that $V=k^{N},P=\sum_{i=1}^{r}x_{i}\tilde{Q}^{i}$ where $\tilde{Q}^{i}$ are quadratic forms. Let $W\subset V$ is defined by the system of equation $\{x_{1}(v)=...=x_{r}(v)=0\}$. We denote elements of $W$ by $y$ and write $v\in V$ in the form $v=(\bar{x},y)$ where $\bar{x}=(x_{1},...,x_{r})$. Then we have

[TABLE]

where $R^{i}$ are polynomials of degree $2$. So there exist quadratic forms $c^{i}$ on $k^{r}$, linear maps $l^{i}\in Hom(k^{r},W^{\vee})$ and quadratic forms $Q^{i}$ on $W$ such that

[TABLE]

We define

[TABLE]

for $\bar{x}\in k^{r}-0$.

For any $\bar{x}\in k^{r}-0$ we define

[TABLE]

By definition we have

[TABLE]

Let $u_{0}=-loq_{q}c(2r+2,3)$ and define

[TABLE]

and denote by $R_{0}\subset k^{r}$ the subspace generated by $U_{0}$. If $U_{0}=\{0\}$ then as before we find that $a_{1}(P)\sim q^{N-r}$ and we see that in this case the conclusion of Theorem 4 is true.

If $R_{0}\neq\{0\}$ then we define $U_{1}$ as earlier in our proof of the case when $r=2$.

4. A proof of Theorem 1

We start notations from section $1.1$ in [D]. For any algebraic $k$-variety $X$ we denote by $\mathcal{D}(X)$ be the bounded derived category of complexes $\mathbb{F}$ of constructible $l$-adic sheaves on $X$. For any $F,G$ in $\mathcal{D}(X)$ we denote by $F\otimes G\in Ob(\mathcal{D}(X))$ the derived tensor product. A morphism $p:X\to Y$ of algebraic varieties defines (derived) functors $p_{\star},p_{!}:\mathcal{D}(X)\to\mathcal{D}(Y)$ and $f^{\star}:\mathcal{D}(Y)\to\mathcal{D}(X)$.

In the case when $Y=\star=Spec(k)$ we can identify cohomology of compexes $p_{\star}(F)$ and $p_{!}(F)$ of $\mathbb{Q}_{l}$-vector spaces with the cohomology groups $H^{\star}(X,F)$ and $H_{c}^{\star}(X,F)$. On the other hand for any point $x\in X(k)$ the stalk $F_{x}$ of $F$ at $x$ is equal to $i_{x}^{\star}(F)$ which is an object of $\mathcal{D}(Spec(k))$ where $i_{x}:Spec(k)\to X$ is the imbedding $x\hookrightarrow X$. We denote by $E_{x}(F)$ the Euler characteristic of the complex $i_{x}^{\star}(F)$.

Let $\psi$ be a non-trivial additive character on $k$ with values in $\bar{\mathbb{Q}}_{l}$ where $l$ is prime number $(q,l)=1$. Deligne defined the Fourier transform functor $\mathcal{F}\in Aut(\mathcal{D})(\mathbb{A})$ by

[TABLE]

where $F$ is an object of $\mathcal{C},p_{1},p_{2},p:\mathbb{A}^{2}\to\mathbb{A}$ maps given by

[TABLE]

(see the section $3$ of [La]) where $\mathbb{A}$ is the affine line over $k$.

The following result follows from results of [La].

Claim 7**.**

If $F\in Ob(\mathcal{D}(\mathbb{A}))$ is $\mathbb{G}_{m}$-equivariant then the Euler characteristic of the stalk $\mathcal{F}(F)_{1}$ at $1\in\mathbb{A}(k)=k$ is equal to the difference $E_{0}(\mathcal{F}(F))-E_{1}(\mathcal{F}(F))$.

One can identify the set of $\mathbb{A}(k)$ with $k$. We define the Frobenious map $Fr_{\mathbb{A}}:\mathbb{A}\to\mathbb{A}$ by $x\to x^{q}$. Then $k$ is the set of fixed points of $Fr_{\mathbb{A}}$. For any $n\geq 1$ we define $\psi_{n}:k_{n}\to\bar{\mathbb{Q}}_{l}$ by $\psi_{n}=\psi\circ tr(k_{n}/k)$ where $k_{n}=\mathbb{F}_{q^{n}}$.

Let $p:\mathbb{A}\to\mathbb{A}$ be the map given by $x\to x^{q}-x$. Then $p:\mathbb{A}\to\mathbb{A}$ is a Galois covering with the group equal to the additive group of $k$ which acts by

[TABLE]

It defines a local system $\mathcal{L}^{AS}$ on $\mathbb{A}$ such that fibers $\mathcal{L}^{AS}_{x}$ of $\mathcal{L}$ at $x\in\mathbb{A}$ are one-dimensional $\bar{\mathbb{Q}}_{l}$-vector spaces. Any morphism

[TABLE]

defines a map

[TABLE]

for any $x\in\mathbb{A}$. In particular for any $x\in k,\alpha_{x}$ is an endomorphism of the line $\mathcal{L}_{x}$. So we can consider it as an element of $\bar{\mathbb{Q}}_{l}$.

Claim 8**.**

There exists unique isomorphism

[TABLE]

such that $\alpha_{x}=\psi(x),x\in k$.

Now let $P:\mathbb{A}^{N}\to\mathbb{A}$ be a polynomial defined over $k,\mathcal{L}_{P}=P^{\star}(\mathcal{L}^{AS})$. Then $\mathcal{L}_{P}$ is a local system on $\mathbb{A}^{N}$ and $\alpha$ defines an isomorphism

[TABLE]

such that

[TABLE]

Let $F=p_{!}(\mathcal{L}_{P}),G=\mathcal{F}(F)$. $\beta$ induces morphisms $\gamma:Fr_{\mathbb{A}}^{\star}(F)\to F$ and $\tilde{\gamma}:Fr_{\mathbb{A}}^{\star}(G)\to G$. We denote by $\gamma_{a},\tilde{\gamma}_{a}$ the induced endomorphisms of complexes $F_{a},G_{a},a\in k$. For any $i$ we denote by $\beta_{i}$ the induced endomorphism of $H^{i}_{c}(\mathcal{A}^{N},\mathcal{L}_{P})$.

The following equality follows from the Lefschetz fixed-point theorem

Claim 9**.**

For any $n\geq 1$ we have

[TABLE]

We define

[TABLE]

As follows from this Claim we have $E(P)=E(G_{1})$.

Lemma 10**.**

If $P$ is homogeneous polynomial of degree $d$ prime to $p$ then $E(P)\neq 0$.

Proof. Since $P$ is homogeneous the object $F$ of $\mathcal{D}(\mathbb{A})$ is $\mathbb{G}_{m}$-equivariant. So the Claims 8 ,9 imply that

[TABLE]

where $X_{a}:=P^{-1}(a)\subset\mathbb{A}^{N}$ and

[TABLE]

for any algebraic $k$-variety $X$.

Since the group $\mathbb{G}_{m}$ acts freely on $X_{0}-\{0\}$ we see that $E(X_{0})=E(\{0\})=1$. On the other hand the group $\mu_{d}$ of $d$-roots of unity acts freely on $\mathcal{A}^{N}-\{0\}$ by

[TABLE]

preserving $X_{1}\subset\mathbb{A}^{N}$. Let $Y=E(X_{1})/\mu_{d}$.

Then $E(X_{1})=dE(Y)\equiv 0$ mod $d$. So $E(P)\neq 0$.

Theorem 1 follows now from Claim 9.

5. A very special case,

In the case when $P=Q^{2}$ where $Q$ is a non-degenerate quadratic form on $V=k^{N}$ we have $|a_{n}(P)|\sim q^{nN/2}$. We see that for quartic polynomials $P$ we can not bound $b(P)$ below in terms of $r(P)$. On the other hand such a bound exists in nice cases.

Lemma 11**.**

5.1 Assume that $P=QR:\mathbb{A}^{N}\to\mathbb{A},deg(Q),deg(R)\neq 0$, that all irreducible components of

[TABLE]

are reducible and defined over $k$ and that $(d,q-1)=1$. There exists a constant $\gamma(d)$ such that

[TABLE]

Proof. Since $(d,q-1)=1$ all the fibers $P^{-1}(a)(k),a\in k^{\star}$ have the same number of elements $A$. It is clear that $A\sim q^{N-1}$. Let $B=|P^{-1}(0)(k)|=|X(k)|$. Then $a_{1}(P)=B-A$. On the other hand $X=Y\cup Z$ where

[TABLE]

So $B=|Y(k)|+|Z(k)|-|Y\cup Z(k)|$ and the Lemma follows from the Weil’s estimates (see [D]).