Related papers: An Algebra Model for the Higher Order Sum Rules
Lienard systems are very important mathematical models describing oscillatory processes arising in applied sciences. In this paper, we study polynomial Lienard systems of arbitrary degree on the plane, and develop a new method to obtain a…
We describe a type system for the linear-algebraic lambda-calculus. The type system accounts for the part of the language emulating linear operators and vectors, i.e. it is able to statically describe the linear combinations of terms…
A well-known theorem by Milnor-Orlik provides a formula for the Milnor number of a weighted-homogeneous polynomial having an isolated singularity that depends only on the weights. In this paper we present a proof of that result using…
We present an algebraic theory of orthogonal polynomials in several variables that includes classical orthogonal polynomials as a special case. Our bottom line is a straightforward connection between apolarity of binary forms and the inner…
Motivated by questions arising in the study of the spectral theory of models of aperiodic order, we investigate sums of functions of semibounded closed subsets of the real line. We show that under suitable thickness assumptions on the sets…
Skew orthogonal polynomials arise in the calculation of the $n$-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular, the distribution functions are completely…
Recent theorems of Dubickas and Mossinghoff use auxiliary polynomials to give lower bounds on the Weil height of an algebraic number $\alpha$ under certain assumptions on $\alpha$. We prove a theorem which introduces an auxiliary polynomial…
We consider multilinear Littlewood polynomials, polynomials in $n$ variables in which a specified set of monomials $U$ have $\pm 1$ coefficients, and all other coefficients are $0$. We provide upper and lower bounds (which are close for $U$…
Let $\mathbf{f} = (f_1, \ldots, f_R)$ be a system of polynomials with integer coefficients in which the degrees need not all be the same. We provide sufficient conditions for which the system of equations $f_j (x_1, \ldots, x_n) = 0 \ (1…
In deduction modulo, a theory is not represented by a set of axioms but by a congruence on propositions modulo which the inference rules of standard deductive systems---such as for instance natural deduction---are applied. Therefore, the…
In this paper, we prove two results related to the solutions of norm form equations. Firstly, we give a finiteness result for sums of terms of linear recurrence sequences appearing in the coordinates of solutions of norm form equations.…
We define and study "tautological classes" in the cohomology of moduli stacks of shtukas, pursuing two directions of applications. First, we prove a formula relating the "arithmetic volume" of tautological classes to higher derivatives of…
This paper is mainly devoted to the study of the differentiation index and the order for quasi-regular implicit ordinary differential algebraic equation (DAE) systems. We give an algebraic definition of the differentiation index and prove a…
For arbitrary $\beta > 0$, we use the orthogonal polynomials techniques developed by R. Killip and I. Nenciu to study certain linear statistics associated with the circular and Jacobi $\beta$ ensembles. We identify the distribution of these…
We will start from the beginning and define a matroid and its Orlik-Solomon algebra and holonomy Lie algebra, but first we give some background from topology and cohomology. A (central) hyperplane arrangement is a finite number of subspaces…
P. Aluffi introduced in [1] a new graded algebra in order to conveniently express characteristic cycles in the theory of singular varieties. This algebra is attached to a surjective ring homomorphism $A\surjects B$ by taking a suitable…
We study volumes of sections of convex origin-symmetric bodies in $\mathbb{R}% ^{n}$ induced by orthonormal systems on probability spaces. The approach is based on volume estimates of \ John-L\"{o}wner ellipsoids and expectations of norms…
We obtain new recurrence relations, an explicit formula, and convolution identities for higher order geometric polynomials. These relations generalize known results for geometric polynomials, and lead to congruences for higher order…
We improve on the first fall degree bound of polynomial systems that arise from a Weil descent along Semaev's summation polynomials relevant to the solution of the Elliptic Curve Discrete Logarithm Problem via Gr\"obner basis algorithms.
This work builds on earlier work of the first three authors where a notion of congruence modules in higher codimension is introduced. The main new results are a criterion for detecting regularity of local rings in terms of congruence…