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Complex Analysis

Edition 1

Dimension-dependent holomorphic certificate degrees on zero-sum hyperplanes

Scott Schubr

Primary subject: Mathematics
Subject subfields: Complex Analysis · Algebra · Linear Algebra and Matrix Theory
Paper type: Research Article
Manuscript stage: Preprint

On the complex zero-sum hyperplane in C^n, consider the symmetric quartics A = n∑ᵢ|vᵢ|⁴ − |∑ᵢvᵢ²|² and Q = ‖v‖⁴ − |∑ᵢvᵢ²|². For every odd n ≥ 3 and every α, β > 0, we prove that a holomorphic squared-norm representation of ‖v‖²ᵐ(αA + βQ) requires m ≥ n − 2. For each fixed odd n, equality holds for all sufficiently small positive β/α, whereas at β = 0 the least exponent is zero. The lower bound uses an explicit Fischer annihilator and a positive Hamming-distance correlation identity on the odd-dimensional sign cube. The upper bound uses the classical quadratic complete intersection associated with the zero-sum hyperplane. For n = 5, we determine the entire first-degree window: the least exponent is three exactly when 0 < β/α ≤ ρ*, where 305ρ*³ + 786ρ*² + 351ρ* − 162 = 0. An explicit four-by-four pencil and harmonic decomposition give this sharp endpoint, including its four-dimensional kernel. We also derive the quartic as a coefficient in a centered permanent deficit, which motivates the family without implying the full permanent inequality.

October 24, 2026Read paper

Classical and Real Analysis

Edition 1

A Sharp Zero-Sector Criterion for Polynomial Prefix Sums

Scott Schubr

Primary subject: Mathematics
Subject subfields: Classical and Real Analysis · Algebra
Paper type: Research Article
Manuscript stage: Preprint

Let h be a real polynomial with h(0)>0 whose zeros lie in the closed sector of half-angle pi/4 around the negative real axis. For every nonnegative real step and geometric bias at least one, its geometrically weighted integer prefix sums admit a finite concave logarithmic extension to the nonnegative quadrant. This implies weighted logarithmic Jensen inequalities for arbitrary probability laws with finite first moments and integral mean prefix length. The universal sector angle and bias range are sharp, including integral-height counterexamples. Joint log-concavity of a two-variable truncated-square lattice sum follows from complete geometry coverage and exact rational certificates; classical Prekopa marginalization removes the degree bound. Applications cover split projective-line bundles over projective space in every dimension. The extension agrees with integer-prefix rows rather than asserting ordinary polynomial continuation in the prefix variable.

October 24, 2026Read paper

Algebra

Edition 1

Concavity of Lattice Counts on Toric Threefolds

Scott Schubr

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

The fourth root of the ordinary Riemann-Roch count polynomial is concave on the entire closed nef cone of every smooth toric Fano threefold. This exponent is uniformly optimal for the ordinary real polynomial extensions. Precisely nine of the eighteen threefolds have concave cube root; the other nine have cube-root failure, certified by exact fan and polynomial computations. Integral nonbig nef classes are included. For split projective-line bundles over the projective plane, the ordinary polynomial has optimal power one third at twists zero, one and two, while no positive concave power exists at any integer twist at least three. Geometrically weighted lattice counts with bias at least one admit nonpolynomial concave logarithmic extensions at every twist, whereas every fixed bias strictly between zero and one has integral midpoint counterexamples at large twists. Ordinary-polynomial optimality and lattice-only sharpness are distinguished.

October 24, 2026Read paper

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.

October 23, 2026Read paper