A proof of the Eisenbud-Huneke-Ulrich conjecture

For general fields the footprint bound from Gröbner basis theory estimates the number of common affine roots of any set of multivariate polynomials using information on their leading monomials. In this paper we develop an interpolation bound with a similar flavor extending a previously known result for only a single polynomial to any prescribed number of polynomials. Surprisingly, our interpolation theorem and the footprint bound can be shown to be two sides of the same coin, solving similar problems, but for dual spaces. As discussed the footprint bound compares well with the improved Alon-Füredi bound and for finite fields the presented interpolation theorem is sharp. Our work can be viewed as a comment to a question raised by Tao in [Tao, 2014]

math.AC↗

arXiv2026

Poincaré Duality and Quadratic Refinements over Laurent Rings

We develop a Poincaré duality theory for defects of nondegenerate sesquilinear pairings over Laurent polynomial rings. A key ingredient is a novel flat resolution of the character module, constructed from a triangulation of the sphere at infinity associated with a fan. The cup product on this resolution turns Poincaré duality on the sphere into canonical pairings between the resulting defect modules. In middle degrees, we construct distinguished quadratic refinements using equivariant cohomology of the sphere with the antipodal action. The effective replacement of the sphere with a projective space provides a geometric substitute for division by two. Applied to translation-invariant Pauli stabilizer codes, our results establish the nondegeneracy of higher-dimensional braiding pairings. They extend the two-dimensional T-junction formula for topological spin to higher dimensions, while giving it a geometric interpretation.

math.AC↗

arXiv2026

Key exchange protocol based on circulant matrix action over congruence-simple semiring

We present a new key exchange protocol based on circulant matrices acting on matrices over a congruence-simple semiring. We describe how to compute matrices with the necessary properties for the implementation of the protocol. Additionally, we provide an analysis of its computational cost and its security against known attacks.

math.AC↗