Short sums of one-variable inhomogeneous polynomials

Prove nontrivial bounds for short character sums with inhomogeneous polynomial arguments in one variable, including the example F(X)=X^2+1.

Background

The paper establishes nontrivial bounds for short sums of Dirichlet characters evaluated at broad classes of multivariable polynomials, including inhomogeneous polynomials when the number of variables is sufficiently large relative to the degree. The authors explicitly identify the one-variable inhomogeneous case as outside the scope of their results.

The stated example F(X)=X2+1 isolates a basic unresolved case in which obtaining nontrivial bounds for short character sums would extend the paper’s multivariable theory to the one-dimensional inhomogeneous setting.

References

Finally, we remark that it still remains an open question to prove nontrivial bounds for short sums of (inhomogeneous) polynomials in one variable, e.g. $F(X)=X2+1$.

— Short character sums of inhomogeneous polynomials  (2609.04092 - Chu, 3 Sep 2026) in Section 1, Introduction