Related papers: Positivity in power series rings

In this note, we give examples of formal power series satisfying certain conditions that cannot be realized as Hilbert series of finitely generated modules. This answers to the negative a question raised in a recent article by the second…

Real algebraic geometry provides certificates for the positivity of polynomials on semi-algebraic sets by expressing them as a suitable combination of sums of squares and the defining inequalitites. We show how Putinar's theorem for…

We give extensions of results on nonnegative matrix semigroups which deduce finiteness or boundedness of such semigroups from the corresponding local properties, e.g., from finiteness or boundedness of values of certain linear functionals…

We consider the set of the power non-negative polynomials of several variables and its subset that consists of polynomials which can be represented as a sum of squares. It is shown in the classic work by D.Hilbert that it is a proper…

This note considers the maximal positively invariant set for polynomial discrete time dynamics subject to constraints specified by a basic semialgebraic set. The note utilizes a relatively direct, but apparently overlooked, fact stating…

Power nonnegative matrices are defined as complex matrices having at least one nonnegative integer power. We exploit the possibility of deriving a Perron Frobenius-like theory for these matrices, obtaining three main results and drawing…

This paper establishes new Positivstellens\"atze for polynomials that are positive on sets defined by polynomial matrix inequalities (PMIs). We extend the classical Handelman and Krivine-Stengle theorems from the scalar inequality setting…

We present a necessary and sufficient condition for a cubic polynomial to be positive for all positive reals. We identify the set where the cubic polynomial is nonnegative but not all positive for all positive reals, and explicitly give the…

We ask whether every polynomial function that is non-negative on a real algebraic curve can be expressed as a sum of squares in the coordinate ring. Scheiderer has classified all irreducible curves for which this is the case. For reducible…

We extend previous results about Putinar's Positivstellensatz for cylinders of type $S \times {\mathbb R}$ to sets of type $S \times {\mathbb R}^r$ in some special cases taking into account $r$ and the degree of the polynomial with respect…

We show the existence of and explicitly construct generic polynomials for various groups, over fields of positive characteristic. The methods we develop apply to a broad class of connected linear algebraic groups defined over finite fields…

The purpose of this paper is to give a characterization for polynomials and rational functions which admit only non-negative values on definable sets in real closed valued fields. That is, generalizing the relative positivstellens\"atze for…

We study the problem of representing multivariate polynomials with rational coefficients, which are nonnegative and strictly positive on finite semialgebraic sets, using rational sums of squares. We focus on the case of finite semialgebraic…

In this article, I introduce a group-theoretical method to prove positivity of certain linear combinations (with coefficients generally lying in $\mathbb{C}$) of exponential functions under a set of semidefinite linear constraints. The…

A long-standing open problem in representation stability is whether every finitely generated commutative algebra in the category of strict polynomial functors satisfies the noetherian property. In this paper, we resolve this problem…

The question of how to certify the non-negativity of a polynomial function lies at the heart of Real Algebra and it also has important applications to Optimization. In the setting of symmetric polynomials Timofte provided a useful way of…

In 2008, M. Marshall settled a long-standing open problem by showing that if f(x,y) is a polynomial that is non-negative on the strip [0,1] x R, then there exist sums of squares s(x,y) and t(x,y) such that f(x,y) = s(x,y) + (x - x^2)…

In this paper we study the representation of Morse polynomial functions which are nonnegative on a compact basic closed semi-algebraic set in $\mathbb R^n$, and having only finitely many zeros in this set. Following C. Bivi\`{a}-Ausina, we…

This paper introduces and develops the algebraic framework of moment polynomials, which are polynomial expressions in commuting variables and their formal mixed moments. Their positivity and optimization over probability measures supported…

A classical problem in real geometry concerns the representation of positive semidefinite elements of a ring $A$ as sums of squares of elements of $A$. If $A$ is an excellent ring of dimension $\geq3$, it is already known that it contains…