arXiv subjects
Publications and source records attributed to Jaka Cimpric.
We show that any finite set of linear partial differential operators with continuous coefficients is linearly dependent if and only if it is locally linearly dependent. It follows that the reflexive closure of any finite set of such operators is equal to its linear span. The last statement can be rephrased as a weak nullstellensatz for linear partial differential operators.
Let $(F,\le)$ be an ordered field and let $A,B$ be square matrices over $F$ of the same size. We say that $A$ and $B$ belong to the same archimedean class if there exists an integer $r$ such that the matrices $r A^T A-B^T B$ and $r B^T B-A^T A$ are positive semidefinite with respect to $\le$. We show that this is true if and only if $A=CB$ for some invertible matrix $C$ such that all entries of $C$ and $C^{-1}$ are bounded by some integer. We also show that every archimedean class contains a row echelon form and that its shape and archimedean classes (in $F$) of its pivots are uniquely determined. For matrices over fields of formal Laurent series we construct a canonical representative in each archimedean class. The set of all archimedean classes is shown to have a natural lattice structure while the semigroup structure does not come from matrix multiplication. Our motivation comes from noncommutative real algebraic geometry and noncommutative valuation theory.
Finsler's Lemma charactrizes all pairs of symmetric $n \times n$ real matrices $A$ and $B$ which satisfy the property that $v^T A v>0$ for every nonzero $v \in \mathbb{R}^n$ such that $v^T B v=0$. We extend this characterization to all symmetric matrices of real multivariate polynomials, but we need an additional assumption that $B$ is negative semidefinite outside some ball. We also give two applications of this result to Noncommutative Real Algebraic Geometry which for $n=1$ reduce to the usual characterizations of positive polynomials on varieties and on compact sets.
Positivity in $\ast$-algebras can be defined either algebraically, by quadratic modules, or analytically, by $\ast$-representations. By the induction procedure for $\ast$-representations we can lift the analytical notion of positivity from a $\ast$-subalgebra to the entire $\ast$-algebra. The aim of this paper is to define and study the induction procedure for quadratic modules. The main question is when a given quadratic module on the $\ast$-algebra is induced from its intersection with the $\ast$-subalgebra. This question is very hard even for the smallest quadratic module (i.e. the set of all sums of hermitian squares) and will be answered only in very special cases.
A method for computing global minima of real multivariate polynomials based on semidefinite programming was developed by N. Z. Shor, J. B. Lasserre and P. A. Parrilo. The aim of this article is to extend a variant of their method to noncommutative symmetric polynomials in variables $X$ and $Y$ satisfying $YX-XY=1$ and $X^\ast=X$, $Y^\ast=-Y$. Global minima of such polynomials are defined and showed to be equal to minima of the spectra of the corresponding differential operators. We also discuss how to exploit sparsity and symmetry. Several numerical experiments are included. The last section explains how our theory fits into the framework of noncommutative real algebraic geometry.
We study non-commutative real algebraic geometry for a unital associative *-algebra A viewing the points as pairs (π,v) where π is an unbounded *-representation of A on an inner product space which contains the vector v. We first consider the *-algebras of matrices of usual and free multivariate polynomials with their natural subsets of points. If all points are allowed then we can obtain results for general A. Finally, we compare our results with their analogues in the usual (i.e. Schmüdgen's) non-commutative real algebraic geometry where the points are unbounded *-representation of A.
Let $A$ be the algebra of all $n \times n$ matrices with entries from $\RR[x_1,\ldots,x_d]$ and let $G_1,\ldots,G_m,F \in A$. We will show that $F(a)v=0$ for every $a \in \RR^d$ and $v \in \RR^n$ such that $G_i(a)v=0$ for all $i$ if and only if $F$ belongs to the smallest real left ideal of $A$ which contains $G_1,\ldots,G_m$. Here a left ideal $J$ of $A$ is real if for every $H_1,\ldots,H_k \in A$ such that $H_1^T H_1+\ldots+H_k^T H_k \in J+J^T$ we have that $H_1,\ldots,H_k \in J$. We call this result the one-sided Real Nullstellensatz for matrix polynomials. We first prove by induction on $n$ that it holds when $G_1,\ldots,G_m,F$ have zeros everywhere except in the first row. This auxiliary result can be formulated as a Real Nullstellensatz for the free module $\RR[x_1,\ldots,x_d]^n$.
We define and study preorderings and orderings on rings of the form $M_n(R)$ where $R$ is a commutative unital ring. We extend the Artin-Lang theorem and Krivine-Stengle Stellensätze (both abstract and geometric) from $R$ to $M_n(R)$. While the orderings of $M_n(R)$ are in one-to-one correspondence with the orderings of $R$, this is not true for preorderings. Therefore, our theory is not Morita equivalent to the classical real algebraic geometry.
This article takes up the challenge of extending the classical Real Nullstellensatz of Dubois and Risler to left ideals in a *-algebra A. After introducing the notions of non-commutative zero sets and real ideals, we develop three themes related to our basic question: does an element p of A having zero set containing the intersection of zero sets of elements from a finite set S of A belong to the smallest real ideal containing S? Firstly, we construct some general theory which shows that if a canonical topological closure of certain objects are permitted, then the answer is yes, while at the purely algebraic level it is no. Secondly for every finite subset S of the free *-algebra R of polynomials in g indeterminates and their formal adjoints, we give an implementable algorithm which computes the smallest real ideal containing S and prove that the algorithm succeeds in a finite number of steps. Lastly we provide examples of noncommutative real ideals for which a purely algebraic non-commutative real Nullstellensatz holds. For instance, this includes the real (left) ideals generated by a finite sets S in the *-algebra of n by n matrices whose entries are polynomials in one-variable. Further, explicit sufficient conditions on a left ideal in R are given which cover all the examples of such ideals of which we are aware and significantly more.
We prove a general archimedean positivstellensatz for hermitian operator-valued polynomials and show that it implies the multivariate Fejer-Riesz Theorem of Dritschel-Rovnyak and positivstellensätze of Ambrozie-Vasilescu and Scherer-Hol. We also obtain several generalizations of these and related results. The proof of the main result depends on an extension of the abstract archimedean positivstellensatz for *-algebras that is interesting in its own right.
For $q>0$ let $\cA$ denote the unital $\ast$-algebra with generator $x$ and defining relation $xx^\ast=qxx^\ast$. Based on this algebra we study $q$-normal operators, the complex $q$-moment problem, positive elements and sums of squares.
We extend Krivine's strict positivstellensatz for usual (real multivariate) polynomials to symmetric matrix polynomials with scalar constraints. The proof is an elementary computation with Schur complements. Analogous extensions of Schm\" udgen's and Putinar's strict positivstellensatz were recently proved by Hol and Scherer using methods from optimization theory.
We present a solution to the real multidimensional rational K-moment problem, where K is defined by finitely many polynomial inequalities. More precisely, let S be a finite set of real polynomials in X=(X_1,...,X_n) such that the corresponding basic closed semialgebraic set K_S is nonempty. Let E=D^{-1}R[X] be a localization of the real polynomial algebra, and T_S^E the preordering on E generated by S. We show that every linear functional L on E that is nonnegative on T_S^E is represented by a positive measure on a certain subset of K_S, provided D contains an element that grows fast enough on K_S.
We consider the problem of determining the closure of a quadratic module M in a commutative R-algebra with respect to the finest locally convex topology. This is of interest in deciding when the moment problem is solvable and in analyzing algorithms for polynomial optimization involving semidefinite programming. The closure of a semiordering is also considered, and it is shown that the space of all semiorderings lying over M plays an important role in understanding the closure of M. The fibre theorem of Schmuedgen for preorderings is strengthened and extended to quadratic modules. The extended result is used to construct an example of a non-archimedean quadratic module describing a compact semialgebraic set that has the strong moment property. The same result is used to obtain a recursive description of the closure of M which is valid in many cases.
We begin a systematic study of positivity and moment problems in an equivariant setting. Given a reductive group $G$ over $\R$ acting on an affine $\R$-variety $V$, we consider the induced dual action on the coordinate ring $\R[V]$ and on the linear dual space of $\R[V]$. In this setting, given an invariant closed semialgebraic subset $K$ of $V(\R)$, we study the problem of representation of invariant nonnegative polynomials on $K$ by invariant sums of squares, and the closely related problem of representation of invariant linear functionals on $\R[V]$ by invariant measures supported on $K$. To this end, we analyse the relation between quadratic modules of $\R[V]$ and associated quadratic modules of the (finitely generated) subring $\R[V]^G$ of invariant polynomials. We apply our results to investigate the finite solvability of an equivariant version of the multidimensional $K$-moment problem. Most of our results are specific to the case where the group $G(\R)$ is compact.
We extend and generalize the results of Scheiderer (2006) on the representation of polynomials nonnegative on two-dimensional basic closed semialgebraic sets. Our extension covers some situations where the defining polynomials do not satisfy the transversality condition. Such situations arise naturally when one considers semialgebraic sets invariant under finite group actions.
We present a new approach to noncommutative real algebraic geometry based on the representation theory of $C^\ast$-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings, \cite[Theorem 5]{jacobi}. We show that this theorem is a consequence of the Gelfand-Naimark representation theorem for commutative $C^\ast$-algebras. A noncommutative version of Gelfand-Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution.
We generalize the notion of and results on maximal proper quadratic modules from commutative unital rings to $\ast$-rings and discuss the relation of this generalization to recent developments in noncommutative real algebraic geometry. The simplest example of a maximal proper quadratic module is the cone of all positive semidefinite complex matrices of a fixed dimension. We show that the support of a maximal proper quadratic module is the symmetric part of a prime $\ast$-ideal, that every maximal proper quadratic module in a Noetherian $\ast$-ring comes from a maximal proper quadratic module in a simple artinian ring with involution and that maximal proper quadratic modules satisfy an intersection theorem. As an application we obtain the following extension of Schm\" udgen's Strict Positivstellensatz for the Weyl algebra: Let $c$ be an element of the Weyl algebra $\mathcal{W}(d)$ which is not negative semidefinite in the Schr\" odinger representation. It is shown that under some conditions there exists an integer $k$ and elements $r_1,...,r_k \in \mathcal{W}(d)$ such that $\sum_{j=1}^k r_j c r_j^\ast$ is a finite sum of hermitian squares. This result is not a proper generalization however because we don't have the bound $k \le d$.