Zero-constant-term Newton-polytope case

Definitions and mathematical statement

For an integer polynomial $f(x_1,\ldots,x_d)=\sum_\alpha a_\alpha x^\alpha$, its support is $\{\alpha:a_\alpha\ne0\}$ and its Newton polytope is the convex hull of this support. The zero-constant-term case is $a_{(0,\ldots,0)}=0$.

Resolve this case in the source's Newton-polytope formulation of multivariate small-root methods. A usual small-root task specifies bounds $B_i$ and seeks integers $r_i$ with $|r_i|\leq B_i$ satisfying either $f(r)=0$ over the integers or $f(r)\equiv0\pmod N$. The record does not identify which root equation, bounds, polynomial class, or lattice construction is intended. The known zero root must not be confused with recovering all relevant nonzero small roots.

The supplied summary says the Jochemsz--May extended strategy has a shift treatment for a zero constant term, sometimes with extra complications or reduced bounds. The intended unresolved aim is a comparably clean treatment preserving the desired small-root range, but no exact theorem or optimality benchmark is given. A translation can change the support and Newton polytope, so merely producing a nonzero constant term is not automatically a solution with unchanged guarantees. No new small-root threshold or unconditional multivariate algorithm is asserted.

Short English statement

Handle zero constant terms in the Newton-polytope small-root framework while preserving the intended guarantees. The actual integer/modular problem, root bounds, and benchmark are missing; returning the trivial zero root does not resolve the stated issue.

Sources

• Alexander May, IPAM Workshop on Number Theory and Cryptography --- Open Problems (2006), attachment 6222, PDF page 25, printed slide 19, 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.
• Jochemsz--May extended strategy, as named in the supplied research summary; no full publication citation or precise shift theorem is provided.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $8800029$, source code catalogue entry, Zero-constant-term Newton-polytope case. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.