Random supports, Newton polytopes, and local Viro complexity

Definitions and mathematical statement

The source asks for connections between "random sparse polygons" and Newton polytopes, extended to random Viro diagrams. It gives no random ensemble, statistic or precise meaning of sparse polygons. In the supplied report's interpretation, a finite exponent set $A\subset{\mathbb{Z}}^d$ has a Bernoulli-selected support $S$; the Newton polytope of a polynomial supported on $S$ is $P=\operatorname{conv}S$. Independent lift heights induce a regular subdivision, and independent fair signs decorate selected vertices. This model is additional to the literal source, not its unique recovered interpretation.

Condition on a pure $d$-dimensional regular triangulation $T$, with $N$ top-dimensional simplices. For each top simplex $\sigma$, its local Viro chart is active when its $d+1$ vertex signs are not all equal. Put $Y=\sum_\sigma I_\sigma$, and let $n_r(T)$ count unordered pairs of distinct top simplices sharing exactly $r$ vertices. The supplied candidate moment formulas are \[ \mathbb E[Y\mid T]=(1-2^{-d})N, \] \[ \operatorname{Var}(Y\mid T)=N(1-2^{-d})2^{-d} +2^{1-2d}\sum_{r=2}^dn_r(T)(2^{r-1}-1). \] Distinct charts sharing zero or one vertex have zero covariance; for $r\ge2$ the reported covariance is $2^{-2d}(2^{r-1}-1)$. The sum is empty for $d=1$. These count active local charts, not connected components or Betti numbers of a global real zero set.

If $b_v$ is the number of top simplices incident to vertex $v$, the report's vertex-incidence McDiarmid estimate has the bounded-differences form \[ \mathbb P(|Y-\mathbb E[Y\mid T]|\ge t\mid T) \le2\exp\left(-\frac{2t^2}{\sum_vb_v^2}\right)\qquad(t>0), \] for a nonempty triangulation. Its exact transfer to random $T$ is the law of total variance, \[ \operatorname{Var}(Y)=\mathbb E[\operatorname{Var}(Y\mid T)] +(1-2^{-d})^2\operatorname{Var}(N(T)). \] This uses a distribution supported on the specified pure full-dimensional triangulations; empty or lower-dimensional support outcomes require separate conventions.

The report also claims an exact directional support-function law for the Bernoulli Newton polytope. With independent inclusion probabilities $p_a$, $h_P(u)=\max_{a\in S}\langle u,a\rangle$, and $h_\varnothing=-\infty$, its standard expression is \[ \mathbb P(h_P(u)\le s)=\prod_{\substack{a\in A\\\langle u,a\rangle>s}}(1-p_a). \] Conditioning on nonempty support changes this distribution. The record does not give the lifting distribution, patchworking hypotheses or a global-topology theorem, so these local and directional results are not promoted to a complete random-Viro classification.

Short English statement

Relate sparse random supports and their Newton polytopes to signed Viro diagrams. Under the supplied independent-support/lifting/sign model, the report gives exact local-chart moments and a directional polytope law. The original ensemble and any global topological conclusion remain unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $20000629$, source code catalogue entry; AIM analysis corpus, aim-analysis-notes.json, notes[160]. Attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0.
• AIM Random analytic functions, April 2006, Question 4 attributed to Maurice Rojas: https://aimath.org/WWN/randomzeros/arcc1.pdf. The literal wording is "random sparse polygons", followed by an extracted control-character artifact. It is not silently rewritten as polynomials. Original remarks and literature fields are empty.
• Registry dated August 14, 2026: statement reconstructed_unverified, status partially_solved, classification PARTIAL-PROGRESS; attempt 1 labelled partial result, rigor proved, novelty confidence low. The Bernoulli/polytope and Viro moment results are machine-generated research aids, not peer reviewed, requiring independent verification before citation. No authoritative external reference is supplied.