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Exponential growth of sparse Euler products and normal graded domains

Scott Schubr

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Abstract

Let d_i be strictly increasing positive integer degrees and let M_i be positive integers with log M_i/d_i→δ∈(0,∞). For the Euler product with M_i independent polynomial generators of degree d_i 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 sup_i log(d_(i+1))/d_i. 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 sup_i d_(i+1)/d_i, 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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  1. Edition 2 · 2026-10-27 · public
  2. Edition 1 · 2026-10-24 · public

Research details

Subjects
Mathematics · Algebra · Combinatorics
Keywords
Euler products, exponential growth, Hilbert functions, graded domains, sparse degree supports
Manuscript date
2026-10-23
Reader reuse
No additional permission granted

Scope

Exact upper and lower exponential coefficient rates for sparse Euler products, contrasted with explicit normal graded domains of dimension two whose exponential Hilbert rates need not converge.

Disclosures

AI systems assisted research exploration, literature discovery, drafting and internal checking. The author retains responsibility for the manuscript. Internal AI-assisted and computational checks are distinct from external human peer review.

Reproduction

6-page manuscript and 5 source-package members. Compile main.tex with a compatible LaTeX engine. The Euler-product rate laws and normal-domain constructions are analytic and require no numerical supplement.

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