Related papers: Low degree polynomial equations: arithmetic, geome…
We present some new and recent algorithmic results concerning polynomial system solving over various rings. In particular, we present some of the best recent bounds on: (a) the complexity of calculating the complex dimension of an algebraic…
This paper surveys a comprehensive, although not exhaustive, sampling of graph polynomials with the goal of providing a brief overview of a variety of techniques defining a graph polynomial and then for decoding the combinatorial…
In this paper we investigate the uniform distribution properties of polynomials in many variables and bounded degree over a fixed finite field F of prime order. Our main result is that a polynomial P : F^n -> F is poorly-distributed only if…
The polynomial hierarchy is a grading of problems by difficulty, including P, NP and coNP as the best known classes. The promise polynomial hierarchy is similar, but extended to include promise problems. It turns out that the promise…
In a classical case, orthogonal polynomial sequences are in such a way that the $ n $th polynomial has the exact degree $n$. Such sequences are complete and form a basis of the space for any arbitrary polynomial. In this paper, we introduce…
Configuration polynomials generalize the classical Kirchhoff polynomial defined by a graph. Their study sheds light on certain polynomials appearing in Feynman integrands. Contact equivalence provides a way to study the associated…
We give a brief survey of recent results on word maps on simple groups and polynomial maps on simple associative and Lie algebras. Our focus is on parallelism between these theories, allowing one to state many new open problems and giving…
The problem of finding all minimal surfaces presented in parametric form as polynomials of certain degree is discussed by many authors. It is known that the classical Enneper surface is (up to position in space and homothety) the only…
Researchers currently use a number of approaches to predict and substantiate information-computation gaps in high-dimensional statistical estimation problems. A prominent approach is to characterize the limits of restricted models of…
Expansive polynomials (whose roots are greater than 1 in modulus) often arise in dynamical systems and other computational problems. This paper examines the expansivity gap (the gap between 1 and the smallest modulus of the roots) of these…
An interesting problem in classical differential geometry is to find methods to prove that two surfaces defined by different charts actually coincide up to position in space. In a previous paper we proposed a method in this direction for…
This article is divided in two parts. In the first part we review some recent results concerning the expected number of real roots of random system of polynomial equations. In the second part we deal with a different problem, namely, the…
Graph polynomials are graph parameters invariant under graph isomorphisms which take values in a polynomial ring with a fixed finite number of indeterminates. We study graph polynomials from a model theoretic point of view. In this paper we…
In this paper, we focus on clarifying the concept of solving equations of degree greater than six using continuous functions or hypergeometric functions and providing another proof of the non-existence of algebraic solutions for equations…
This paper discusses the split feasibility problem with polynomials. The sets are semi-algebraic, defined by polynomial inequalities. They can be either convex or nonconvex, either feasible or infeasible. We give semidefinite relaxations…
In a previous work of the authors, a result to algorithmically compute the topology types of the level curves of an algebraic surface, is given. From this result, here we derive applications based on level curves to determine some…
Graph polynomials are polynomials assigned to graphs. Interestingly, they also arise in many areas outside graph theory as well. Many properties of graph polynomials have been widely studied. In this paper, we survey some results on the…
In recent years, the notion of characteristic polynomial of representations of Lie algebras has been widely studied. This paper provides more properties of these characteristic polynomials. For simple Lie algebras, we characterize the…
We introduce the general polynomial algebras characterizing a class of higher order superintegrable systems that separate in Cartesian coordinates. The construction relies on underlying polynomial Heisenberg algebras and their defining…
Fine-grained complexity theory is the area of theoretical computer science that proves conditional lower bounds based on the Strong Exponential Time Hypothesis and similar conjectures. This area has been thriving in the last decade, leading…