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In this paper an equation means a homogeneous linear partial differential equation in $n$ unknown functions of $m$ variables which has real or complex polynomial coefficients. The solution set consists of all $n$-tuples of real or complex analytic functions that satisfy the equation. For a given system of equations we would like to characterize its Weyl closure, i.e. the set of all equations that vanish on the solution set of the given system. It is well-known that in many special cases the Weyl closure is equal to $B_m(F)N \cap A_m(F)^n$ where $F$ is either the field of real or complex numbers, $A_m(F)$ (respectively $B_m(F)$) consists of all linear partial differential operators with coefficients in $F[x_1,\ldots,x_m]$ (respectively $F(x_1,\ldots,x_m)$) and $N$ is the submodule of $A_m(F)^n$ generated by the given system. Our main result is that this formula holds in general. In particular, we do not assume that the module $A_m(F)^n/N$ has finite rank which used to be a standard assumption. Our approach works also for the real case which was not possible with previous methods. Moreover, our proof is constructive as it depends only on the Riquier-Janet theory., Comment: 10 pages, submitted

For every associative algebra $A$ and every class $\mathcal{C}$ of representations of $A$ the following question (related to nullstellensatz) makes sense: Characterize all tuples of elements $a_1,\ldots,a_n \in A$ such that vectors $\pi(a_1)v,\ldots,\pi(a_n)v$ are linearly dependent for every $\pi \in \mathcal{C}$ and every $v$ from the representation space of $\pi$. We answer this question in the following cases: (1) $A=U(L)$ is the enveloping algebra of a finite-dimensional complex Lie algebra $L$ and $\mathbb{C}$ is the class of all finite-dimensional representations of $A$. (2) $A=U(\mathfrak{sl}_2(\mathbb{C}))$ and $\mathbb{C}$ is the class of all finite-dimensional irreducible representations of $A$. (3) $A=U(\mathfrak{sl}_3(\mathbb{C}))$ and $\mathbb{C}$ is the class of all finite-dimensional irreducible representations of $A$ with sufficiently high weights. In case (1) the answer is: tuples that are linearly dependent over $\mathbb{C}$ while in cases (2) and (3) the answer is: tuples that are linearly dependent over the center of $A$. Similar results have been proved before for free algebras and Weyl algebras., Comment: 21 pages

Positivity in *-algebras can be defined either algebraically, by quadratic modules, or analytically, by *-representations. By the induction procedure for *-representations, we can lift the analytical notion of positivity from a *-subalgebra to the entire *-algebra. The aim in this article is to define and study the induction procedure for quadratic modules. The main question is when a given quadratic module on the *-algebra is induced from its intersection with the *-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.

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., 23 pages, 2 figures, submitted

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., 7 pages, v2

For q > 0 q>0 let A \mathcal {A} denote the unital ∗ * -algebra with generator x x and defining relation x x ∗ = q x ∗ x xx^*=qx^*x . Based on this algebra we study q q -normal operators and the complex q q -moment problem. Among other things, we prove a spectral theorem for q q -normal operators, a variant of Haviland’s theorem and a strict Positivstellensatz for A . \mathcal {A}. We also construct an example of a positive element of A \mathcal {A} which is not a sum of squares. It is used to prove the existence of a formally q q -normal operator which is not extendable to a q q -normal one in a larger Hilbert space and of a positive functional on A \mathcal {A} which is not strongly positive.

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 Schmudgen’s and Putinar’s strict positivstellensatz were recently proved by Hol and Scherer using methods from optimization theory.

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., 12 pages

Every total ordering of a commutative domain can be extended uniquely to its field of fractions. This result is extended in two directions. Firstly, the notion of a total ordering is generalized so that a nonzero element can have more than two signs (in fact, these signs form a group). Secondly, commutative domains are replaced by noncommutative ones and we consider the following types of rings of fractions: Ore extensions, maximal (right or two-sided) rings of fractions, division hulls of free algebras and epic fields. Throughout the paper several examples are given to illustrate the theory.

For every domain R and every even integer n, we define ms n (R) (resp. ps n (R)) as the smallest number k such that zero is a sum of k + 1 products (resp. permuted products) of nth powers of nonzero elements from R. There are many results about ps n in the literature but nothing is known about ms n . We prove two results about ms n of twisted Laurent series rings R((x,ω)). The first result is that if ms 2 (R) = ∞ and ω has order n/ 2 in Aut(R), then ms n (R((x,ω))) = ∞. The second result is that there exist R and ω such that ms n (R((x,ω))) = ∞ and ps n (R((x,ω)))

The n -th product level of a skew–field D , ps n ( D ) , is a generalization of the n -th level of a field F , s n ( F ) . An explicit bound for s 2 m ( F ) in terms of m and s 2 ( F ) is known and it is also known that there is no such bound for ps 2 m ( D ) when m is even. Our aim is to explicitly construct such a bound for odd m . More precisely, we construct a function f : N 3 → N , such that ps 2 k l ( D ) ⩽ f ( ps 2 k ( D ) , k , l ) , for every integer k , every odd integer l and every skew–field D . We give an explicit bound for the n -th Pythagorean number of a simple extension Q ( d ) / Q in Section 2 and prove a noncommutative version of Hilbert identities in Section 3. These two results are used in Section 4 in the proof of our main result. In section 5, we show that the n -th product level of an Ore domain is equal to the n -th product level of its skew field of fractions.

The aim of this paper is to introduce the notion of the nth product level ps n of an associative unital ring and study its properties. Our main results are that every Noetherian ring A with ps n (A) < ∞ for some n has ps nl (A) < ∞ for every odd number l (Theorem 8) and that for every even n there exists a skew-field D with ps n (D) = 1 and ps 2n (D) = ∞ (Theorem 9). This is in a sharp contrast with the commutative case. Namely, by Proposition 4.6 in [1], for every commutative unital ring R with ps 2 (R) < ∞ we have that ps n (R) < ∞ for every n .

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 $\RR[x_1,\ldots,x_d]^n$., v1 7 pages. v2 9 pages: revised abstract, extended introduction and references. To appear in J. Algebra

Haviland’s theorem states that given a closed subset K in R n , each functional L : R [ x ¯ ] → R positive on Pos ( K ) ≔ { p ∈ R [ x ¯ ] | p | K ≥ 0 } admits an integral representation by a positive Borel measure. Schmudgen proved that in the case of compact semialgebraic set K it suffices to check positivity of L on a preordering T , having K as the non-negativity set. Further he showed that the compactness of K is equivalent to the archimedianity of T . The aim of this paper is to extend these results from functionals on the usual real polynomials to operators mapping from the real matrix or operator polynomials into R , M n ( R ) or B ( K ) .

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 Stellensatze (both abstract and geometric) from R to M n ( R ) . This problem has been open since the seventies when Hilbertʼs 17th problem was extended from usual to matrix polynomials. 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.

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., Comment: 10 pages, 1 figure

We consider the problem of determining the closure \(\bar M\) 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 [28][29] and in analyzing algorithms for polynomial optimization involving semidefinite programming [12]. The closure of a semiordering is also considered, and it is shown that the space \(\mathcal{Y}_M\) consisting of all semiorderings lying over M plays an important role in understanding the closure of M. The result of Schmudgen for preorderings in [29] 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 \(\bar M\) which is valid in many 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., Comment: The title updated

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 $k \le d$., 11 pages

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., Comment: 28 pages, 3 figures

An involution $#$ on an associative ring $R$ is \textit{formally real} if a sum of nonzero elements of the form $r^# r$ where $r \in R$ is nonzero. Suppose that $R$ is a central simple algebra (i.e. $R=M_n(D)$ for some integer $n$ and central division algebra $D$) and $#$ is an involution on $R$ of the form $r^# = a^{-1} r^\ast a$, where $\ast$ is some transpose involution on $R$ and $a$ is an invertible matrix such that $a^\ast=\pm a$. In section 1 we characterize formal reality of $#$ in terms of $a$ and $\ast|_D$. In later sections we apply this result to the study of formal reality of involutions on crossed product division algebras. We can characterize involutions on $D=(K/F,\Phi)$ that extend to a formally real involution on the split algebra $D \otimes_F K \cong M_n(K)$. Every such involution is formally real but we show that there exist formally real involutions on $D$ which are not of this form. In particular, there exists a formally real involution $#$ for which the hermitian trace form $x \mapsto \tr(x^#x)$ is not positive semidefinite., Comment: 16 pages

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 positivstellensatze 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.