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2 papers · Page 1 of 1

Combinatorics

Edition 1

Effective coefficient reserves for homogeneous real-stable polynomials

Scott Schubr

Primary subject
Mathematics
Subject subfields
Combinatorics
Paper type
Research Article
Manuscript stage
Preprint

For homogeneous real-stable polynomials of degree n in n variables, with nonnegative coefficients and value one at the all-ones vector, we give explicit positive rational constants in every fixed degree for a global prescribed-gradient coefficient reserve. On an explicitly specified closed neighborhood of the balanced gradient, the squarefree coefficient deficit dominates both centered-root dispersion and the coefficient ℓ¹ distance from the corresponding power of a linear form. The proof replaces the compactness step in the earlier qualitative theorem by a radial extension to a degree-drop boundary and a positive contraction dichotomy. An explicit positive map then corrects the gradient while preserving the centered factors. The bounds include singular dispersion and stable coefficient boundaries. We also obtain an explicit Frobenius-distance reserve for Hermitian mixed discriminants. The constants are deliberately coarse; no optimal, dimension-independent or algorithmic bound is asserted.

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    Algebra

    Edition 1

    Exponential growth of sparse Euler products and normal graded domains

    Scott Schubr

    Primary subject
    Mathematics
    Subject subfields
    Algebra · Combinatorics
    Paper type
    Research Article
    Manuscript stage
    Preprint

    Let dᵢ be strictly increasing positive integer degrees and let Mᵢ be positive integers with log(Mᵢ)/dᵢ → δ ∈ (0, ∞ ). For the Euler product with Mᵢ independent polynomial generators of degree dᵢ and one degree-one generator, we prove that the upper logarithmic coefficient rate is δ and the lower rate is max{δ − θ, 0}, where θ = lim supᵢ log(dᵢ₊₁)/dᵢ. This gives an exact convergence criterion with no regularity assumption on the support gaps. We contrast it with normal connected locally finite graded domains of Krull dimension two, obtained by adapting an infinite-pole construction of Maclean. Their lower rate is δ/R, where R = lim supᵢ dᵢ₊₁/dᵢ, and their upper rate is δ. In particular, normality, finite dimension and a degree-one nonzerodivisor do not jointly force convergence of the exponential Hilbert rate.

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