Related papers: An Algebra Model for the Higher Order Sum Rules
Let X be a finite set of points in R^n. A polynomial p nonnegative on X can be written as a sum of squares of rational functions modulo the vanishing ideal I(X). From the point of view of applications, such as polynomial optimization, we…
This is an introduction to algebraic combinatorics, written for a quarter-long graduate course. It starts with a rigorous introduction to formal power series with some combinatorial applications, then discusses integer partitions (proving…
Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be…
The paper investigates the problem of performing correlation analysis when the number of observations is very large. In such a case, it is often necessary to combine the random observations to achieve dimensionality reduction of the…
Many problems in combinatorial linear algebra require upper bounds on the number of solutions to an underdetermined system of linear equations $Ax = b$, where the coordinates of the vector $x$ are restricted to take values in some small…
We introduce an elementary method to study the border rank of polynomials and tensors, analogous to the apolarity lemma. This can be used to describe the border rank of all cases uniformly, including those very special ones that resisted a…
We provide a new foundational approach to the generalization of terms up to equational theories. We interpret generalization problems in a universal-algebraic setting making a key use of projective and exact algebras in the variety…
In this paper we describe various applications of the Riemann-Hilbert method to the theory of orthogonal polynomials on the line and on the circle.
We construct a combinatorial generalization of the Leray models for hyperplane arrangement complements. Given a matroid and some combinatorial blowup data, we give a presentation for a bigraded (commutative) differential-graded algebra. If…
We obtain new combinatorial formulae for modified Hall--Littlewood polynomials, for matrix elements of the transition matrix between the elementary symmetric functions and Hall-Littlewood's ones, and for the number of rational points over…
This paper introduces an analogue of the Solomon descent algebra for the complex reflection groups of type $G(r,1,n)$. As with the Solomon descent algebra, our algebra has a basis given by sums of `distinguished' coset representatives for…
We evaluate in closed form several classes of finite trigonometric sums. Two general methods are used. The first is new and involves sums of roots of unity. The second uses contour integration and extends a previous method used by two of…
We present a Bounded Model Checking technique for higher-order programs. The vehicle of our study is a higher-order calculus with general references. Our technique is a symbolic state syntactical translation based on SMT solvers, adapted to…
High-order reconstruction schemes for the solution of hyperbolic conservation laws in orthogonal curvilinear coordinates are revised in the finite volume approach. The formulation employs a piecewise polynomial approximation to the…
It is believed that Dirichlet series with a functional equation and Euler product of a particular form are associated to holomorphic newforms on a Hecke congruence group. We perform computer algebra experiments which find that in certain…
We study generalized sums of linear orders. These are binary operations that, given linear orders $A$ and $B$, return an order $A \oplus B$ that can be decomposed as an isomorphic copy of $A$ interleaved with a copy of $B$. We show that…
We develop a theory of arithmetic Newton polygons of higher order, that provides the factorization of a separable polynomial over a $p$-adic field, together with relevant arithmetic information about the fields generated by the irreducible…
We use derived Hall algebra of the category of nilpotent representations of Jordan quiver to reconstruct the theory of symmetric functions, focusing on Hall-Littlewood symmetric functions and various operators acting on them.
Using the theory of noncommutative symmetric functions, we introduce the higher order peak algebras, a sequence of graded Hopf algebras which contain the descent algebra and the usual peak algebra as initial cases (N = 1 and N = 2). We…
In this paper we define a class of polynomial functors suited for constructing coalgebras representing processes in which uncertainty plays an important role. In these polynomial functors we include upper and lower probability measures,…