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I'm trying to figure out how to approach the following question:

Let $g,h$ be polynomials over $\mathbb{Z}_p$ (for prime $p$) with $n>1$ variables. Denote by $bias(g)=|\sum_{x\in \mathbb{Z}_p^n}e^{2\pi i \cdot g(x)/p}|$ (and equivalently for $h$).

Suppose I know that $bias(g)\ne bias(h)$. Find a (non-trivial) lower bound on $|bias(g)- bias(h)|$.

Another variant for this problem is the following:

Given two vectors $v_g,v_h\in \mathbb{Z}_p^n$ that represent the histogram of these polynomials (a list of $p$ entries that specified how many times the polynomial evaluated to each element in the field), give (non-trivial) upper bound for $||v_g-v_h||_1$ given that the $v_g\ne v_h$.

This problem seems to have a "flavour" of something like the Schwartz-Zippel lemma, in the sense that (it may be that) "polynomials with different histogram/bias have very different histogram/bias" (in comparison to "non-identical polynomials are non-equal most of the time" of the Schwartz-Zippel lemma), but it seems to require different techniques. In particular, I don't think the Weil/Deligne bounds for exponential sums helps here.

Thanks!

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    $\begingroup$ You can make polynomials take on any values you want, so this is just asking about sums of $p^n$ $p$-th roots of unity. $\endgroup$ Nov 17 '20 at 23:01
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if accept square it several times, then use vanderport trick actrually we can have a good expension of $|\operatorname{bias}(g)|^{2^{deg(g)}}$, where $\operatorname{bias}(g)=\left|\sum_{x \in Z_{p}^{n}} e^{2 \pi i \cdot g(x) / p}\right|$, from this maybe we can gain some nontrivial lower estimate for $|\operatorname{bias}(g)-\operatorname{bias}(h)|$, if $g\neq h$. Just a example to expain what happen when $deg(g)=2, g(x)=x^2+cx+d$,

$\begin{aligned} |\operatorname{bias}(g)|^2&=\operatorname{bias}(g) \overline{\operatorname{bias}(g)}\\ &=|\sum_{x \in Z_{p}^{n}} \sum_{y \in Z_{p}^{n}} e^{2 \pi(g(x)-y(y)) / p}| \\ &=|\sum_{x \in Z_{p}^{n}} \sum_{a \in Z_{p}^{n}} e^{2 \pi(g(x+a)-g(x)) / p}|\\&=|\sum_{a \in Z_{p}^{n}}\sum_{x \in Z_{p}^{n}} e^{2 \pi(2ax+a^2+ca) / p}| \end{aligned}$

and with the last one we can get some nontrivial estimate for lowerbound. In fact the toy model problem for $deg(f)=deg(g)=2$ wiil be find a lower bound for, $$min_{c_1,c_2}(|\sum_{a \in Z_{p}^{n}}\sum_{x \in Z_{p}^{n}} e^{2 \pi(2ax+a^2+c_1a) / p}|-|\sum_{a \in Z_{p}^{n}}\sum_{x \in Z_{p}^{n}} e^{2 \pi(2ax+a^2+c_2a) / p}|)$$

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