Related papers: A Symbolic Homotopy Algorithm for Solving Composab…
The algebra of holomorphic polynomial Sp_{2n}-invariants of k complex 2n by 2n matrices (under diagonal conjugation action) is generated by the traces of words in these matrices and their symplectic adjoints. No concrete minimal generating…
We are concerned with the problem of decomposing the parameter space of a parametric system of polynomial equations, and possibly some polynomial inequality constraints, with respect to the number of real solutions that the system attains.…
Efficient characteristic set methods for computing solutions of polynomial equation systems in a finite field are proposed. The concept of proper triangular sets is introduced and an explicit formula for the number of solutions of a proper…
We present an effective method for computing parametric primary decomposition via comprehensive Gr\"obner systems. In general, it is very difficult to compute a parametric primary decomposition of a given ideal in the polynomial ring with…
We consider the problem of computing exact sums of squares (SOS) decompositions for certain classes of non-negative multivariate polynomials, relying on semidefinite programming (SDP) solvers. We provide a hybrid numeric-symbolic algorithm…
We present an algorithm to compute the action of the Hecke operators on the top dimensional integral cohomology of certain torsion-free arithmetic subgroups of algebraic groups of Q-rank one. This generalizes the modular symbol algorithm to…
This paper presents a novel algorithm for constructing a sum-of-squares (SOS) decomposition for positive semi-definite polynomials with rational coefficients. Unlike previous methods that typically yield SOS decompositions with…
Homological algebra techniques can be found in almost all modern areas of mathematics. Many interesting problems in mathematics can be formulated, computed, or can find their equivalence in terms of Ext-groups. For instance, important…
As showed in (Fiedler, 1990), any polynomial can be expressed as a characteristic polynomial of a complex symmetric arrowhead matrix. This expression is not unique. If the polynomial is real with only real distinct roots, the matrix can be…
Analytic combinatorics studies asymptotic properties of families of combinatorial objects using complex analysis on their generating functions. In their reference book on the subject, Flajolet and Sedgewick describe a general approach that…
$ \ell_1 $-regularized linear inverse problems are frequently used in signal processing, image analysis, and statistics. The correct choice of the regularization parameter $ t \in \mathbb{R}_{\geq 0} $ is a delicate issue. Instead of…
For every number field and every Cartan Killing type, there is an associated split simple algebraic group. We examine whether the corresponding arithmetic subgroups are profinitely solitary so that the commensurability class of the…
Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of…
This paper studies the polynomial optimization problem whose feasible set is a union of several basic closed semialgebraic sets. We propose a unified hierarchy of Moment-SOS relaxations to solve it globally. Under some assumptions, we prove…
We extend the symbolic representation to the ring of N=1 supersymmetric differential polynomials, and demonstrate that operations on the ring, such as the super derivative, Frechet derivative and super commutator, can be carried out in the…
We prove a complexity dichotomy for complex-weighted Holant problems with an arbitrary set of symmetric constraint functions on Boolean variables. This dichotomy is specifically to answer the question: Is the FKT algorithm under a…
Endowing the set of functional graphs (FGs) with the sum (disjoint union of graphs) and product (standard direct product on graphs) operations induces on FGs a structure of a commutative semiring R. The operations on R can be naturally…
Given a polynomial map $f:\Bbb C^{n+1}\to\Bbb C$, one can attach to it a geometrical variation of mixed Hodge structures (MHS) which gives rise to a limit MHS. The equivariant Hodge numbers of this MHS are analytical invariants of the…
Let $X$ and $Y$ be finite simplicial sets (e.g. finite simplicial complexes), both equipped with a free simplicial action of a finite group $G$. Assuming that $Y$ is $d$-connected and $\dim X\le 2d$, for some $d\geq 1$, we provide an…
Let $ \mathcal{A}_1, \ldots, \mathcal{A}_k $ be finite sets in $ \mathbb{Z}^n $ and let $ Y \subset (\mathbb{C}^*)^n $ be an algebraic variety defined by a system of equations \[ f_1 = \ldots = f_k = 0, \] where $ f_1, \ldots, f_k $ are…