Bounds On Exponential Sums With Quadrinomials

TL;DR

This paper improves bounds on exponential sums involving quadrinomials over finite fields, particularly focusing on sums over multiplicative subgroups, advancing the understanding of exponential sum estimates.

Contribution

The paper introduces a new bound on exponential sums with quadrinomials by refining existing results on exponential quadrilinear sums over multiplicative subgroups.

Findings

Enhanced bounds on exponential sums with quadrinomials

Improved estimates for sums over multiplicative subgroups

Advancement in exponential sum analysis techniques

Abstract

We improve an existing result on exponential quadrilinear sums in the case of sums over multiplicative subgroups of a finite field and use it to give a new bound on exponential sums with quadrinomials.

Equations

$$\Psi (X)=\sum _{i=1}^t{a}_i{X}^{k_i}$$

$$S_{\chi }(\Psi )=\sum _{x\in {\mathbb{F}}_p^{\ast }}\chi (x){\mathbf{e}}_p(\Psi (x)),$$

$$S_{\chi }(\Psi )\le \max \{k_1,\dots ,k_t\}{p}^{1/2},$$

$$\Psi (X)=a{X}^k+b{X}^{\ell }+c{X}^m+d{X}^n.$$

$$\Psi (X)=a{X}^k+f(X^{\ell })+g(X^m)+h(X^n)$$

$$S_{\chi }(\Psi )\ll {\left(\frac{k\ell mn}{\max (k,\ell ,m,n)}\right)}^{1/9}{p}^{8/9}$$

$$\frac{k\ell mn}{\max (k,\ell ,m,n)}<p,$$

$$S_{\chi }(\Psi )\ll (k\ell mn{)}^{1/16}{p}^{7/8}$$

$$\alpha =\gcd (k,p-1),\beta =\gcd (\ell ,p-1),\gamma =\gcd (m,p-1),\delta =\gcd (n,p-1)$$

$$f=\frac{\alpha }{\gcd (\alpha ,\delta )},g=\frac{\beta }{\gcd (\beta ,\delta )},h=\frac{\gamma }{\gcd (\gamma ,\delta )}.$$

$$S_{\chi }(\Psi )\ll$$

$$\displaystyle+\left\{\begin{array}[]{ll}p^{15/16}\delta^{1/32},&\text{if $g\geq p^{1/2}\log p$},\\ p^{31/32}\delta^{1/32}g^{-1/16+o(1)},&\text{if $f\geq p^{1/2}\log p>g$},\\ p\delta^{1/32}(fg)^{-1/16+o(1)},&\text{if $p/\delta\geq p^{1/2}\log p>f$},\\ p^{31/32+o(1)}\delta^{3/32}(fg)^{-1/16},&\text{if $p/\delta<p^{1/2}\log p$}.\end{array}\right.$$

$$\sum _{x\in \mathcal{X}}|{\alpha }_x{|}^2=A\text{mbox}and\sum _{y\in \mathcal{Y}}|{\beta }_y{|}^2=B,$$

$$\left|\sum _{x\in \mathcal{X}}\sum _{y\in \mathcal{Y}}{\alpha }_x{\beta }_y{\mathbf{e}}_p(xy)\right|\le \sqrt{pAB}.$$

$$(u_1-v_1)(u_2-v_2)=(u_3-v_3)(u_4-v_4),u_i,v_i\in \mathcal{U},i=1,2,3,4.$$

$$u_1{v}_1=u_2{v}_2{u}_i\in \mathcal{U},v_i\in \mathcal{V},i=1,2.$$

$$D_{\times}({\mathcal{G}})\ll\left\{\begin{array}[]{ll}|{\mathcal{G}}|^{8}p^{-1},&\text{if $|{\mathcal{G}}|\geq p^{1/2}\log p$},\\ |{\mathcal{G}}|^{6}\log|{\mathcal{G}}|,&\text{if $|{\mathcal{G}}|<p^{1/2}\log p$}.\end{array}\right.$$

$${\mathrm{E}}^{\times}({\mathcal{G}}+\lambda)-\frac{|{\mathcal{G}}|^{4}}{p}\ll\left\{\begin{array}[]{ll}p^{1/2}|{\mathcal{G}}|^{3/2},&\text{if $|{\mathcal{G}}|\geq p^{2/3}$},\\ |{\mathcal{G}}|^{3}p^{-1/2},&\text{if $p^{2/3}>|{\mathcal{G}}|\geq p^{1/2}\log p$},\\ |{\mathcal{G}}|^{2}\log|{\mathcal{G}}|,&\text{if $|{\mathcal{G}}|<p^{1/2}\log p$}.\end{array}\right.$$

$${\mathrm{E}}^{\times}({\mathcal{G}}+\lambda)\ll\left\{\begin{array}[]{ll}|{\mathcal{G}}^{4}|/p,&\text{if $|{\mathcal{G}}|\geq p^{1/2}\log p$},\\ |{\mathcal{G}}|^{2}\log|{\mathcal{G}}|,&\text{if $|{\mathcal{G}}|<p^{1/2}\log p$}.\end{array}\right.$$

$$\displaystyle N({\mathcal{F}},{\mathcal{G}},{\mathcal{H}})\ll\frac{F^{2}}{M^{1/2}}\left\{\begin{array}[]{ll}G^{2}H^{2}p^{-1/2},&\text{if $H\geq p^{1/2}\log p$},\\ G^{2}H^{3/2+o(1)}p^{-1/4},&\text{if $G\geq p^{1/2}\log p>H$,}\\ (GH)^{3/2+o(1)},&\text{if $G<p^{1/2}\log p$.}\end{array}\right.$$

$$N(\mathcal{F},\mathcal{G},\mathcal{H})$$

$$N(\mathcal{F},\mathcal{G},\mathcal{H}{)}^2$$

$$=\frac{F^4{G}^2{H}^2}{|S|}({\mathrm{E}}^{\times }(\mathcal{G}-1){\mathrm{E}}^{\times }(\mathcal{H}-1){)}^{1/2}.$$

$$\displaystyle N({\mathcal{F}},{\mathcal{G}},{\mathcal{H}})^{2}\ll\frac{F^{4}G^{2}H^{2}}{|S|}\left\{\begin{array}[]{ll}G^{2}H^{2}/p,&\text{if $H\geq p^{1/2}\log p$},\\ G^{2}H^{1+o(1)}p^{-1/2},&\text{if $G\geq p^{1/2}\log p>H$,}\\ (GH)^{1+o(1)},&\text{if $G<p^{1/2}\log p$.}\end{array}\right.$$

$$\underset{(w,x,y)\in \mathcal{W}\times \mathcal{X}\times \mathcal{Y}}{\max }|{\vartheta }_{w,x,y}|\le 1,\underset{(w,x,y)\in \mathcal{W}\times \mathcal{X}\times \mathcal{Z}}{\max }|{\rho }_{w,x,z}|\le 1,$$

$$\underset{(w,x,y)\in \mathcal{W}\times \mathcal{Y}\times \mathcal{Z}}{\max }|{\sigma }_{w,y,z}|\le 1,\underset{(w,x,y)\in \mathcal{X}\times \mathcal{Y}\times \mathcal{Z}}{\max }|{\tau }_{x,y,z}|\le 1,$$

$$T=\sum _{w\in \mathcal{W}}\sum _{x\in \mathcal{X}}\sum _{y\in \mathcal{Y}}\sum _{z\in \mathcal{Z}}{\vartheta }_{w,x,y}{\rho }_{w,x,z}{\sigma }_{w,y,z}{\tau }_{x,y,z}{\mathbf{e}}_p(awxyz)$$

$$|T|$$

$$\displaystyle\qquad+\left\{\begin{array}[]{ll}W^{31/32}XYZp^{-1/32},&\text{if $Y\geq p^{1/2}\log p$},\\ W^{31/32}XY^{15/16+o(1)}Z,&\text{if $X\geq p^{1/2}\log p>Y$},\\ W^{31/32}(XY)^{15/16+o(1)}Zp^{1/32},&\text{if $W\geq p^{1/2}\log p>X$},\\ W^{29/32+o(1)}(XY)^{15/16}Zp^{1/16},&\text{if $W<p^{1/2}\log p$}.\end{array}\right.$$

$$|T{|}^8\ll (WXY{)}^6{Z}^7\sum _{\mu \in {\mathbb{F}}_p^{\ast }}\sum _{\lambda \in {\mathbb{F}}_p}J(\mu )I(\lambda ){\eta }_{\mu }{\mathbf{e}}_p(\lambda \mu )+(WXZ{)}^8{Y}^7,$$

$$\displaystyle\sum_{\mu\in{\mathbb{F}}^{*}_{p}}J(\mu)^{2}\ll\left\{\begin{array}[]{ll}X^{4}Y^{4}/p,&\text{if $Y\geq p^{1/2}\log p$},\\ X^{4}Y^{3+o(1)}p^{-1/2},&\text{if $X\geq p^{1/2}\log p>Y$},\\ (XY)^{3+o(1)},&\text{if $X<p^{1/2}\log p$}.\end{array}\right.$$

$$\sum _{\lambda \in {\mathbb{F}}_p}I(\lambda {)}^2$$

Full text

1. Introduction

1.1. Set Up

For a prime $p$, we use ${\mathbb{F}}_{p}$ to denote the finite field of $p$ elements.

For a $t$-sparse polynomial

[TABLE]

with some pairwise distinct non-zero integer exponents $k_{1},\ldots,k_{t}$ and coefficients $a_{1},\ldots,a_{t}\in{\mathbb{F}}_{p}^{*}$, and a multiplicative character $\chi$ of ${\mathbb{F}}_{p}^{*}$ we define the sums

[TABLE]

where ${\mathbf{\,e}}_{p}(u)=\exp(2\pi iu/p)$ and $\chi$ is an arbitrary multiplicative character of ${\mathbb{F}}_{p}^{*}$. The challenge for such sums is to provide a bound that is stronger than the Weil bound

[TABLE]

see [18, Appendix 5, Example 12], by taking advantage of the arithmetic structure of the exponents. The case of exponential sums of monomials has seen much study with Shparlinski [16] providing the first such bound. Further improvements have been made by various other authors, see [3, 1, 11, 12, 15, 17]. We also mention that Cochrane, Coffelt and Pinner, as well as others, have given several bounds on exponential sums with sparse polynomials, see [4, 5, 6, 7, 8, 9] and references therein, some of which we outline in Section 1.2.

Here we provide some new bounds on quadrinomial exponential sums using the techniques in [13]. We thus define

[TABLE]

We mention that all our results extend naturally to more general sums with polynomials of the shape

[TABLE]

for polynomials $f,g,h\in{\mathbb{F}}_{p}[X]$.

The notation $A\ll B$ is equivalent to $|A|\leq c|B|$ for some constant $c$.

1.2. Previous Results

We compare our result for quadrinomials (1.1) to those of Cochrane, Coffelt and Pinner [4, Theorem 1.1]

[TABLE]

which is non-trivial for

[TABLE]

and of Cochrane and Pinner [6, Theorem 1.1]

[TABLE]

which is non-trivial for $k\ell mn<p^{2}$. Our new result in Theorem 1.1 is independent of the size of the exponents but instead depends on various greatest common divisors.

1.3. Main Result

Our main result is the following theorem.

Theorem 1.1**.**

Let $\Psi(X)$ be a quadrinomial of the form (1.1) with $a,b,c,d\in{\mathbb{F}}_{p}^{*}$. Define

[TABLE]

and

[TABLE]

Suppose $f\geq g\geq h$, then $p/\delta\geq f$ and

[TABLE]

We mention that our result is independent of the size of our powers $k,l,m,n$ and is strongest when $\delta$ is small and $f,g,h$ are large. As mentioned in the previous section, previous results become trivial for quadrinomials of large degree. It is easy to see that our bound is non-trivial and improves previous results for a wide range of exponents $k,\ell,m$ and $n$.

2. Preliminaries

We recall the following classical bound of bilinear sums, see, for example, [2, Equation 1.4] or [10, Lemma 4.1].

Lemma 2.1**.**

For any sets ${\mathcal{X}},{\mathcal{Y}}\subseteq{\mathbb{F}}_{p}$ and any $\alpha=(\alpha_{x})_{x\in{\mathcal{X}}}$, $\beta=\left(\beta_{y}\right)_{y\in{\mathcal{Y}}}$, with

[TABLE]

we have

[TABLE]

We define $D_{\times}({\mathcal{U}})$ to be the number of solutions of

[TABLE]

We also define the multiplicative energy ${\mathrm{E}}^{\times}({\mathcal{U}},{\mathcal{V}})$ to be the number of solutions of

[TABLE]

When ${\mathcal{U}}={\mathcal{V}}$, we write ${\mathrm{E}}^{\times}({\mathcal{U}},{\mathcal{U}})=E^{\times}({\mathcal{U}})$.

We need the following result from [13, Corollary 3.3].

Lemma 2.2**.**

For a multiplicative subgroup ${\mathcal{G}}\subset{\mathbb{F}}_{p}^{*}$, we have

[TABLE]

We also use [13, Corollary 4.1].

Lemma 2.3**.**

Let ${\mathcal{G}}$ be a multiplicative subgroup of ${\mathbb{F}}_{p}^{*}$. Then for any $\lambda\in{\mathbb{F}}_{p}^{*}$, we have

[TABLE]

We immediately obtain the following result by observing the dominant term from Lemma 2.3.

Corollary 2.4**.**

Let ${\mathcal{G}}$ be a multiplicative subgroup of ${\mathbb{F}}_{p}^{*}$. Then for any $\lambda\in{\mathbb{F}}_{p}^{*}$, we have

[TABLE]

We define $N({\mathcal{F}},{\mathcal{G}},{\mathcal{H}})$ to be the number of triples of solutions to $f_{1}(g_{1}-g_{2})=f_{2}(h_{1}-h_{2})$ where $f_{i}\in{\mathcal{F}},g_{i}\in{\mathcal{G}},h_{i}\in{\mathcal{H}}$ for $i=1,2$. Using Corollary 2.4 we obtain the following result.

Lemma 2.5**.**

Let ${\mathcal{F}},{\mathcal{G}},{\mathcal{H}}$ be multiplicative subgroups of ${\mathbb{F}}^{*}_{p}$ with cardinalities $F,G,H$ respectively with $G\geq H$. Additionally, let $M=\max(F,G)$. Then

[TABLE]

Proof.

By multiplying both sides of $f_{1}(g_{1}-g_{2})=f_{2}(h_{1}-h_{2})$ by the inverses $f_{2}^{-1}$ and $h_{2}^{-1}$ and taking a factor of $g_{2}$ from the left hand side, and defining $S=\{fgh:f\in{\mathcal{F}},g\in{\mathcal{G}},h\in{\mathcal{G}}\}$ we have

[TABLE]

By the Cauchy inequality,

[TABLE]

By Corollary 2.4,

[TABLE]

Since $|S|\geq M$ we complete our proof.     \sqcap$$\sqcup

Applying Lemma 2.2 and Lemma 2.5 in the proof of [14, Theorem 1.4], we obtain the following result on quadrilinear sums over subgroups.

Lemma 2.6**.**

For any multiplicative subgroups ${\mathcal{W}},{\mathcal{X}},{\mathcal{Y}},{\mathcal{Z}}\subseteq{\mathbb{F}}_{p}^{*}$ of cardinalities $W,X,Y,Z$, respectively, with $W\geq X\geq Y\geq Z$ and weights $\vartheta=(\vartheta_{w,x,y})$, $\rho=(\rho_{w,x,z})$, $\sigma=(\sigma_{w,y,z})$ and $\tau=(\tau_{x,y,z})$ with

[TABLE]

[TABLE]

for the sums

[TABLE]

we have

[TABLE]

uniformly over $a\in{\mathbb{F}}_{p}^{*}$.

Proof.

We see from [14, p. 24] that

[TABLE]

where $\eta_{\mu}$, $\mu\in{\mathbb{F}}^{*}_{p}$ is a complex number with $|\eta_{\mu}|=1$, $J(\mu)$ is the number of quadruples $(x_{1},x_{2},y_{1},y_{2})\in{\mathcal{X}}^{2}\times{\mathcal{Y}}^{2}$ such that $(x_{1}-x_{2})(y_{1}-y_{2})=\mu\in{\mathbb{F}}^{*}_{p}$ and $I(\lambda)$ is the number of triples $(w_{1},w_{2},z)\in{\mathcal{W}}^{2}\times{\mathcal{Z}}$ such that $z(w_{1}-w_{2})=\lambda\in{\mathbb{F}}_{p}$. We estimate $J(\mu)$ as in [14, Equation 3.10] but using our bound from Lemma 2.2 to obtain

[TABLE]

Now

[TABLE]

Therefore, by Lemma 2.5,

[TABLE]

Applying the classical bound on bilinear exponential sums from Lemma 2.1 together with (2.4) and (2.7), we get

[TABLE]

Hence,

[TABLE]

This completes the proof.     \sqcap$$\sqcup

We compare our bound for subgroups from Lemma 2.6 with that for arbitrary sets coming from [14, Theorem 1.4]

[TABLE]

For example, if $W=X=Y=Z=p^{1/2+o(1)}$ then the bounds become $p^{125/64+o(1)}$ and $p^{63/32+o(1)}$ respectively.

3. Proof of Theorem 1.1

Let ${\mathcal{G}}_{\alpha},{\mathcal{G}}_{\beta},{\mathcal{G}}_{\gamma}$ be the subgroups of ${\mathbb{F}}^{*}_{p}$ formed by the elements of orders $\alpha,\beta$ and $\gamma$ respectively. Then,

[TABLE]

where $\vartheta_{w,x,y}=\chi(wxy){\mathbf{\,e}}_{p}(cw^{m}x^{m}y^{m})$, $\rho_{w,x,z}=\chi(z){\mathbf{\,e}}_{p}(bw^{\ell}x^{\ell}z^{\ell})$ and $\sigma_{w,y,z}={\mathbf{\,e}}_{p}(aw^{k}y^{k}z^{k})$. Now the image ${\mathcal{W}}=\{w^{n}:w\in{\mathbb{F}}^{*}_{p}\}$ of non-zero $n$th powers contains $(p-1)/\delta$ elements, each appearing with multiplicity $\delta$. Similarly, we can see that the images ${\mathcal{X}}=\{x^{n}:x\in{\mathcal{G}}_{\alpha}\},{\mathcal{Y}}=\{y^{n}:y\in{\mathcal{G}}_{\beta}\}$ and ${\mathcal{Z}}=\{z^{n}:z\in{\mathcal{G}}_{\gamma}\}$ contain $f,g$ and $h$ elements with multiplicity $\gcd(\alpha,\delta),\gcd(\beta,\delta)$ and $\gcd(\gamma,\delta)$ respectively. We apply Lemma 2.6, recalling our assumption that $f\geq g$ and noticing $f\delta={\mathrm{lcm}}(\alpha,\delta)<p-1$, hence $f\leq p/\delta$, which gives us

[TABLE]

This concludes the proof.