Let f∈Z[x,y], let C_p be the curve modulo p defined by f(x,y)equiv0 pmod p, and write e_p^2(u)=exp(2 pi i u/p^2). Exponential-sum conjecture. There exists a constant D>0 such that, for every prime p and every k,l∈0,1,…,p-1, ∑_(x,y)∈ C_pe_p^2bigl(f(x+kp,y+lp)bigr)≤ Dp^1/2. This is proposed as an exponential-sum estimate that may be obtainable by methods analogous to Bombieri's work and could be used to study the standard discrepancy conjecture. The supplied text gives no resolution.

Exponential-sum conjecture for shifted polynomial values modulo p squared

About 2 years old · traced to 2024

Let $f\in\mathbb{Z}[x,y]$ , let $C_p$ be the curve modulo $p$ defined by $f(x,y)\equiv0\pmod p$ , and write $e_{p^2}(u)=\exp(2\pi i u/p^2)$ . Exponential-sum conjecture. There exists a constant $D>0$ such that, for every prime $p$ and every $k,l\in\{0,1,\ldots,p-1\}$ ,

\sum_{(x,y)\in C_p}e_{p^2}\bigl(f(x+kp,y+lp)\bigr)\leq Dp^{1/2}.

This is proposed as an exponential-sum estimate that may be obtainable by methods analogous to Bombieri's work and could be used to study the standard discrepancy conjecture. The supplied text gives no resolution.

References

Primary source

Krishnan Rajkumar and Shubham, “On the Distribution of Points of Valuation 1 for a Polynomial in Two Variables”, arXiv:2411.05366 (2026).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A reader-written calculation claims the conjecture is false even for a straight line, but nobody has independently verified it.

Rajkumar and Shubham proposed a uniform square-root estimate for the shifted values of a polynomial on its curve modulo $p$ , with shifts $k,l\in\{0,\ldots,p-1\}$ ; the preprint was first posted in 2024. The paper presents this estimate as a conjectural input for a Poisson-distribution result, not as a theorem.

Posted attempt

The calculation takes $f(x,y)=x+y+1$ , for which $C_p=\{(x,p-1-x):0\le x<p\}$ . It claims the exponential sum is exactly $p\,e_p(1+k+l)$ , hence has magnitude $p$ , contradicting any bound of order $p^{1/2}$ for unbounded primes. This is presented as a complete counterexample, but it has not been independently verified.

Current status (as of August 2026): The primary paper leaves the conjecture open, while a complete counterexample has been claimed in reader-written material and remains unverified.

Sources

arxiv.org

CounterexampleThis solution needs a summarySee full solutionHide full solution

Conjecture 4 is false, even for a smooth, absolutely irreducible affine line. Take the fixed polynomial

f(x,y)=x+y+1.

Its two partial derivatives are both $1$ . Thus the simultaneous congruences $f=f_x=f_y=0$ have no solution modulo any prime, as required by the standing hypothesis in Section 3 of the paper.

Let $p$ be any prime. Use the paper's canonical representatives $0\le x,y<p$ , and let $C_p$ be the pairs among them satisfying $p\mid f(x,y)$ . Since

1\le x+y+1\le 2p-1,

the divisibility condition is equivalent to $x+y+1=p$ . Consequently

\begin{gathered} C_p=\{(x,p-1-x):0\le x<p\},\\ |C_p|=p. \end{gathered}

Write $e_m(u)=\exp(2\pi i u/m)$ . For $0\le k,l<p$ , denote by $S_p(k,l)$ the sum of $e_{p^2}(f(x+kp,y+lp))$ over $(x,y)\in C_p$ . At every point of $C_p$ ,

\begin{aligned} f(x+kp,y+lp) &=x+y+1+p(k+l)\\ &=p(1+k+l). \end{aligned}

All $p$ summands are therefore equal, giving the exact identity

S_p(k,l)=p\,e_p(1+k+l), \qquad |S_p(k,l)|=p.

The additive character $e_{p^2}$ is primitive, not trivial. In particular, at the allowed zero shift every summand is $e_p(1)\ne1$ . The same obstruction remains at the nonzero shift $(k,l)=(1,0)$ for every odd prime: every summand is $e_p(2)\ne1$ .

For this single fixed polynomial, a bound $|S_p(k,l)|\le D\sqrt p$ with $D$ independent of $p$ would force $\sqrt p\le D$ for every prime. This is impossible because primes are unbounded. Thus the conjectured square-root bound fails, even if $D$ is allowed to depend on $f$ .

The conclusion does not depend on supplying absolute-value bars to the printed inequality. The permitted nonzero shift $(k,l)=(p-1,0)$ gives $f(x+kp,y+lp)=p^2$ at every point of $C_p$ , and hence the real positive sum $S_p(p-1,0)=p$ , which also exceeds $D\sqrt p$ for primes $p>D^2$ .

The representative convention is on page 3, the standing hypothesis is on page 12, and the conjecture is on page 16 of Rajkumar and Shubham, On the Distribution of Points of Valuation 1 for a Polynomial in Two Variables, Integers 26 (2026), A82.