Related papers: Parallels between quaternionic and matrix nullstel…

We introduce the Z-polynomial of a matroid, which we define in terms of the Kazhdan-Lusztig polynomial. We then exploit a symmetry of the Z-polynomial to derive a new recursion for Kazhdan-Lusztig coefficients. We solve this recursion,…

We present novel equivalences in random matrix and tensor models between complex and self-adjoint theories with nontrivial quadratic terms in the action, established through an intermediate field representation. More precisely, we show that…

The main result of this note is a tracial Nullstellensatz for free noncommutative polynomials evaluated at tuples of matrices of all sizes: Suppose f_1,...,f_r,f are free polynomials, and tr(f) vanishes whenever all tr(f_j) vanish. Then…

This paper is concerned with the factorization and equivalence problems of multivariate polynomial matrices. We present some new criteria for the existence of matrix factorizations for a class of multivariate polynomial matrices, and obtain…

In this short note we give an elementary proof of the fact that connections and their geometric parallel-transport counterpart are equivalent notions.

We compile a long list of equivalent formulations of Hilbert's Nullstellensatz in infinite dimensions, and prove a persistence result for the strong Nullstellensatz in large polynomial rings.

We give an alternative proof for the equivalence of two definitions of the totally positive grassmannian.

We settle three problems from the literature on stable and real zero polynomials and their connection to matroid theory. We disprove the weak real zero amalgamation conjecture by Schweighofer and the second author. We disprove a conjecture…

We complete the proof of the Howe duality conjecture in the theory of local theta correspondence by treating the remaining case of quaternionic dual pairs in arbitrary residual characteristic.

We give an overview of our effort to introduce (dual) semicanonical bases in the setting of symmetrizable Cartan matrices.

The present paper continues our foundational work on real algebra with preordered commutative semifields and semirings. We prove two abstract Vergleichsstellens\"atze for preordered commutative semirings of polynomial growth. These…

In this paper we prove a strong version of the Hilbert Nullstellensatz in the ring $\mathbb H[q_1,\ldots,q_n]$ of slice regular polynomials in several quaternionic variables. Our proof deeply depends on a detailed analysis of the common…

This article extends the classical Real Nullstellensatz to matrices of polynomials in a free $\ast$-algebra $\RR\axs$ with $x=(x_1, \ldots, x_n)$. This result is a generalization of a result of Cimpri\vc, Helton, McCullough, and the author.…

In the paper [1] we consider a new class, so-called, $G$-monogenic (differentiable in the sense of Gateaux) quaternionic mappings. In the present paper we introduce quaternionic $H$-monogenic (differentiable in the sense of Hausdorff)…

We discuss several existing proofs of the value of a quartic integral and present a new proof that evolved from rational Landen transformations.

Wei's celebrated Duality Theorem is generalized in several ways, expressed as duality theorems for linear codes over division rings and, more generally, duality theorems for matroids. These results are further generalized, resulting in two…

We establish a factorisation theorem for invertible, cross-symmetric, totally nonnegative matrices, and illustrate the theory by verifying that certain cases of Holte's Amazing Matrix are totally nonnegative.

Recent status of neutrino oscillation phenomenology with four neutrinos is reviewed. It is emphasized that the so-called (2+2)-scheme as well as the (3+1)-scheme are still consistent with the recent solar and atmospheric neutrino data.

We survey on the recent progress toward mirror symmetry between Landau-Ginzburg models.

We establish new explicit connections between classical (scalar) and matrix Gegenbauer polynomials, which result in new symmetries of the latter and further give access to several properties that have been out of reach before: generating…