Related papers: B\'ezoutians and the $\mathbb{A}^1$-degree

For each configuration of rational points on the affine line, we define an operation on the group of unstable A1 motivic homotopy classes of endomorphisms of the projective line. We also derive an algebraic formula for the image of such an…

We introduce the degree and local degree in equivariant motivic homotopy theory for the purpose of studying equivariant enumerative problems over general fields. Given a finite, tame group scheme $G$ over a field $k$ and an equivariant…

The classical version of B\'ezout's Theorem gives an integer-valued count of the intersection points of hypersurfaces in projective space over an algebraically closed field. Using work of Kass and Wickelgren, we prove a version of…

In this sequel to arxiv:arXiv:1012.0835 we develop Bezout type theorems for semidegrees (including an explicit formula for {\em iterated semidegrees}) and an inequality for subdegrees. In addition we prove (in case of surfaces) a Bernstein…

We establish an algebra-isomorphism between the complexified Grothendieck ring F of certain bimodule categories over a modular tensor category and the endomorphism algebra of appropriate morphism spaces of those bimodule categories. This…

We use discrete Morse theory to give a new proof of the Degree Theorem in Auter space A_n. There is a filtration of A_n into subspaces A_{n,k} using the degree of a graph, and the Degree Theorem says that each A_{n,k} is (k-1)-connected.…

This paper gives an elementary proof of an improved version of the algebraic Local B\'ezout Theorem (given by the authors in JSC 45 (2010) 975--985). Here we remove some ad hoc hypotheses and obtain an optimal algebraic version of the…

We prove that the local $\mathbb{A}^1$-degree of a polynomial function at an isolated zero with finite separable residue field is given by the trace of the local $\mathbb{A}^1$-degree over the residue field. This fact was originally…

We consider topological conditions under which a locally invertible map admits a global inverse. Our main theorem states that a local diffeomorphism $f: M \to\mathbb{R}^n$ is bijective if and only if $H_{n-1}(M)=0$ and the pre-image of…

We prove a formula for the multidegrees of a rational map defined by generalized monomials on a projective variety, in terms of integrals over an associated Newton region. This formula leads to an expression of the multidegrees as volumes…

Let $E$ be an elliptic curve over a number field $K$. If for almost all primes of $K$, the reduction of $E$ modulo that prime has rational cyclic isogeny of fixed degree, we can ask if this forces $E$ to have a cyclic isogeny of that degree…

Recently S. Patrikis, J.F. Voloch and Y. Zarhin have proven, assuming several well known conjectures, that the finite descent obstruction holds on the moduli space of principally polarised abelian varieties. We show an analogous result for…

Let $A$ and $B$ be abelian varieties defined over the function field $k(S)$ of a smooth algebraic variety $S/k.$ We establish criteria, in terms of restriction maps to subvarieties of $S,$ for existence of various important classes of…

We establish ring isomorphisms between quantum Grothendieck rings of certain remarkable monoidal categories of finite-dimensional representations of quantum affine algebras of types $A_{2n-1}^{(1)}$ and $B_n^{(1)}$. Our proof relies in part…

One can compute the local $\mathbb{A}^1$-degree at points with separable residue field by base changing, working rationally, and post-composing with the field trace. We show that for endomorphisms of the affine line, one can compute the…

The article is a contribution to the local theory of geometric Langlands correspondence. The main result is a categorification of the isomorphism between the (extended) affine Hecke algebra, thought of as an algebra of Iwahori bi-invariant…

This dissertation comprises three collections of results, all united by a common theme. The theme is the study of categories via algebraic techniques, considering categories themselves as algebraic objects. This algebraic approach to…

We are interested in classifying groups of local biholomorphisms (or even formal diffeomorphisms) that can be endowed with a canonical structure of algebraic group up to add extra formal diffeomorphisms. We show that this is the case for…

We construct an exact tensor functor from the category $\mathcal{A}$ of finite-dimensional graded modules over the quiver Hecke algebra of type $A_\infty$ to the category $\mathscr C_{B^{(1)}_n}$ of finite-dimensional integrable modules…

We prove a monoidal equivalence between spectral and automorphic realizations of the universal affine Hecke category, thereby proving the tamely ramified local Betti geometric Langlands correspondence, as conjectured by Ben-Zvi--Nadler…