Powers of the Vandermonde determinant are eventually non-SNP

Abstract

We prove a conjecture of Monical, Tokcan, and Yong that every fixed positive power of the Vandermonde determinant is non-SNP in all sufficiently many variables, where a polynomial is non-SNP if there is a lattice point in its Newton polytope that does not appear with nonzero coefficient. This means our result proves that for every even power $k\geq4$, there is always such a missing lattice monomial in large enough dimensions. The odd case follows from alternation, and the quadratic case was previously known. For every even power $k\geq4$, we exhibit an explicit lattice point in the Newton polytope of $a_{δ_k}^k$ whose coefficient vanishes. The vanishing is obtained from a Dyson constant-term identity, proved using the finite-variable Jack scalar product and Macdonald's specialization formula. The key even-power construction and proof strategy arose from prompting with OpenAI Codex (GPT Sol 5.6 Extra High), a large language model; the complete transcript appears in the appendix. The authors subsequently checked and organized the argument. The accompanying Lean formalization is available at https://github.com/steven-le-thien/vandermonde-snp.

Disclosure

“= (−1)r (r + 1)tr . r≥0 Comparing coefficients of tr yields (29): (73) ϵ−1/m (Er ) = (−1)r (r + 1). 7. G ENERATIVE AI USAGE OpenAI Codex (GPT Sol 5.6 Extra High) was used as an interactive research and writing aid in developing this proof. All prompts and follow-up instructions to Codex were sup- plied by the first author. In the initial conversation (reproduced below)”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T19:09:31.544361Z

Structural counts