Edition 3
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Combinatorics and Graph Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
We prove a global sixth-root Jensen inequality for bounded unimodular fibers whose central affine dimension is at most three. Every integrable integer-valued demand with feasible fibers almost surely and integral mean satisfies the inequality when the central fiber is bounded and has affine dimension at most three, including countable support. A monotone concave extension of the sixth root of every type-A3 alcoved lattice count allows all twelve directed bounds to vary. Classical chamber formulas and Lorentzian volume theory combine with degree-six homogenization, wall gluing, shortest-path closure and a common integral rank-three reduction. Stronger planar powers, K4 extension sharpness and integral cube-root counterexamples are retained. Directed rounding and a triangle obstruction distinguish local methods from global Jensen. The general sixth-root exponent is not claimed optimal; unrestricted higher-dimensional unimodular Jensen remains open.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Combinatorics and Graph Theory · Probability and Statistics
- Paper type
- Research Article
- Manuscript stage
- Preprint
For the unrooted spanning-forest measure on any finite simple graph with arbitrary positive edge weights, the full incident-edge marginal at every vertex is real stable. Each star is strongly Rayleigh, incident edges are nonpositively correlated, and vertex degrees have Poisson-binomial laws. The proof combines a classical optional-component-mark forest polynomial with a stable determinant differential operator and a signed matching polynomial. We also prove strict pairwise negative edge correlation on every complete graph with at least three vertices and every complete bipartite graph with both parts at least two, at every common positive fugacity. Exact Hermite and Laguerre counts, continued fractions, scalar crossing barriers and polynomial certificates treat disjoint edges in these families. Quantitative bipartite critical and supercritical deficit asymptotics are retained. Arbitrary-host disjoint-edge correlation and negative association of the full forest law remain outside the results.
Edition 2
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Combinatorics and Graph Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
For each fixed degree n, we prove that among homogeneous real-stable polynomials in n variables with nonnegative coefficients, value one at the all-ones vector, and a prescribed gradient sufficiently close to the all-ones vector, the power of the corresponding linear form uniquely minimizes the squarefree coefficient. This answers Question 1.8 of Gurvits–Leake, arXiv:2609.21083v2. We strengthen the minimum statement to a uniform lower bound for the coefficient deficit by the coefficient ℓ₁ distance to that power. The distance order is optimal. We also classify strict local minimality at every positive gradient and give a cubic example separating local and global minimization. The proof uses centered real-root dispersion, an exact coefficient identity and the classical balanced uniqueness theorem. The neighborhood and constants are dimension-dependent and qualitative. We also give a total-variation interpretation for stable count distributions and an illustrative nondeterminantal quartic family.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Combinatorics and Graph Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
For uniform linear extensions of finite posets with bounded maximum incomparability degree, a local stable-partition estimate tracks the incomparability budget consumed by conditioning. The resulting three-vertex order bound gives a sharper explicit bounded-range balance estimate through the established finite-boundary argument. A complementary two-chain construction quantifies a rare-triple obstruction to reusing the sharp pair denominator. Complete proofs and exact finite reproducibility code accompany these counting refinements.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Combinatorics and Graph Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
Choose uniformly a nonempty tree subgraph of the labeled complete graph, including single-vertex trees. For every n ≥ 3 and two distinct adjacent edges, we prove that the conditional probability of one edge, given the other, exceeds three quarters of its unconditional probability. The short proof combines classical fixed-forest extension counts, Jensen’s inequality and a count of four-vertex trees. The constant is sharp by the previously established asymptotic limit.