Nondegenerate surfaces in toric threefolds

Definitions and mathematical statement

In the finite-field setting suggested by the workshop, let $S/\mathbb F_q$ be a surface in a three-dimensional toric variety. Its zeta function is \[ Z(S,t)=\exp\left(\sum_{r\geq1}\frac{\#S(\mathbb F_{q^r})}{r}t^r\right). \] Develop effective algorithms computing this rational function exactly for the source's nondegenerate surfaces in toric threefolds.

A standard hypersurface setting starts with a Laurent polynomial $f\in\mathbb F_q[x_1^{\pm1},x_2^{\pm1},x_3^{\pm1}]$ and its Newton polytope $\Delta$, the convex hull of its exponent vectors. For a nonempty face $\tau\subseteq\Delta$, let $f_\tau$ be the sum of terms on that face. Newton nondegeneracy means that \[ f_\tau=x_1\frac{\partial f_\tau}{\partial x_1} =x_2\frac{\partial f_\tau}{\partial x_2} =x_3\frac{\partial f_\tau}{\partial x_3}=0 \] has no solution in $(\overline{\mathbb F}_q^*)^3$ for every such face, including $\Delta$. This gives a conventional meaning of nondegenerate, not an independently verified definition from the slide.

The input/output theorem should specify whether $S$ is the open torus hypersurface or its compactification in a toric threefold, the fan and smoothness hypotheses, and the exact required notion of effectiveness. The two zeta functions differ by boundary contributions. The source supplies no complexity target in $q$, characteristic, degrees, Newton-polytope size, or input length, so no polynomial-time bound is invented.

Short English statement

Compute exact zeta functions effectively for nondegenerate surfaces in toric threefolds over finite fields. Fix the nondegeneracy and compactification conventions and state the algorithm's input representation and cost guarantees.

Sources

• Kiran Kedlaya, IPAM Workshop on Number Theory and Cryptography --- Open Problems (2006), attachment 6220, PDF pages 40--41, logical slide 21, schedule source https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule; accessed 2026-07-29. The supplied provenance records a PDF/legacy-PPT visual audit.
• The research summary mentions work attributed to Castryck--Chtcherbakov and Gajovic on toric/projective surfaces and higher-dimensional toric $p$-adic algorithms, but supplies no article titles, identifiers, or complete theorem hypotheses.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $8800017$, source code catalogue entry, Nondegenerate surfaces in toric threefolds. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.