We investigate quadrature rules for measures supported on real algebraic and rational curves, focusing on the odd-degree case 2 s − 1 . Adopting an optimization viewpoint, we minimize suitable penalty functions over the space of quadrature rules of strength 2 s − 1 , so that optimal solutions yield rules with the minimal number of nodes. For plane algebraic curves of degree d , we derive explicit node bounds depending on d and the number of places at infinity, improving results of Riener–Schweighofer, and Zalar. For rational curves in arbitrary dimension of degree d , we further refine these bounds using the geometry of the parametrization and recover the classical Gaussian quadrature bound when d = 1 . Our results reveal a direct link between the algebraic complexity of the supporting curve and the minimal size of quadrature formulas, providing a unified framework that connects real algebraic geometry, polynomial optimization, and moment theory.

Let G be a finite group acting linearly on R n . A celebrated Theorem of Procesi and Schwarz gives an explicit description of the orbit space R n / / G as a basic closed semi-algebraic set. We give a new proof of this statement and another description as a basic closed semi-algebraic set using elementary tools from real algebraic geometry. Bröcker was able to show that the number of inequalities needed to describe the orbit space generically depends only on the group G . Here, we construct such inequalities explicitly for abelian groups and in the case where only one inequality is needed. Furthermore, we answer an open question raised by Bröcker concerning the genericity of his result.

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The classes of sums of arithmetic-geometric exponentials (SAGE) and of sums of nonnegative circuit polynomials (SONC) provide nonnegativity certificates which are based on the inequality of the arithmetic and geometric means. We study the cones of symmetric SAGE and SONC forms and their relations to the underlying symmetric nonnegative cone. As main results, we provide several symmetric cases where the SAGE or SONC property coincides with nonnegativity and we present quantitative results on the differences in various situations. The results rely on characterizations of the zeroes and the minimizers for symmetric SAGE and SONC forms, which we develop. Finally, we also study symmetric monomial mean inequalities and apply SONC certificates to establish a generalized version of Muirhead's inequality.

A semi-algebraic set is a subset of the real space defined by polynomial equations and inequalities. In this paper, we consider the problem of deciding whether two given points in a semi-algebraic set are connected. We restrict to the case when all equations and inequalities are invariant under the action of the symmetric group and of degree at most d < n, where n is the number of variables. Additionally, we assume that the two points are in the same fundamental domain of the action of the symmetric group, by assuming that the coordinates of two given points are sorted in non-decreasing order. We construct and analyze an algorithm that solves this problem, by taking advantage of the group action, and has a complexity being polynomial in n.

In this paper, we consider the problem of deciding the existence of real solutions to a system of polynomial equations having real coefficients, and which are invariant under the action of the symmetric group. We construct and analyze a Monte Carlo probabilistic algorithm which solves this problem, under some regularity assumptions on the input, by taking advantage of the symmetry invariance property.

We consider cones of real forms which are sums of squares and invariant under a (finite) reflection group. Using the representation theory of these groups we are able to use the symmetry inherent in these cones to give more efficient descriptions. We focus especially on the An, Bn, and Dn case where we use so-called higher Specht polynomials to give a uniform description of these cones. These descriptions allow us, to deduce that the description of the cones of sums of squares of fixed degree 2d stabilizes with n>2d. Furthermore, in cases of small degree, we are able to analyze these cones more explicitly and compare them to the cones of non-negative forms.

Abstract We study symmetric non-negative forms and their relationship with symmetric sums of squares. For a fixed number of variablesnand degree 2d, symmetric non-negative forms and symmetric sums of squares form closed, convex cones in the vector space ofn-variate symmetric forms of degree 2d. Using representation theory of the symmetric group we characterize both cones in a uniform way. Further, we investigate the asymptotic behavior when the degree 2dis fixed and the number of variablesngrows. Here, we show that, in sharp contrast to the general case, the difference between symmetric non-negative forms and sums of squares does not grow arbitrarily large for any fixed degree 2d. We consider the case of symmetric quartic forms in more detail and give a complete characterization of quartic symmetric sums of squares. Furthermore, we show that in degree 4 the cones of non-negative symmetric forms and symmetric sums of squares approach the same limit, thus these two cones asymptotically become closer as the number of variables grows. We conjecture that this is true in arbitrary degree 2d.

An ideal of polynomials is symmetric if it is closed under permutations of variables. We relate general symmetric ideals to the so called Specht ideals generated by all Specht polynomials of a given shape. We show a connection between the leading monomials of polynomials in the ideal and the Specht polynomials contained in the ideal. This provides applications in several contexts. Most notably, this connection gives information about the solutions of the corresponding set of equations. From another perspective, it restricts the isotypic decomposition of the ideal viewed as a representation of the symmetric group.

Let R be a real closed field. We consider basic semi-algebraic sets defined by n -variate equations/inequalities of s symmetric polynomials and an equivariant family of polynomials, all of them of degree bounded by 2d < n. Such a semi-algebraic set is invariant by the action of the symmetric group. We show that such a set is either empty or it contains a point with at most 2d-1 distinct coordinates. Combining this geometric result with efficient algorithms for real root finding (based on the critical point method), one can decide the emptiness of basic semi-algebraic sets defined by s polynomials of degree d in time (sn)O(d). This improves the state-of-the-art which is exponential in n . When the variables x1, łdots, xn are quantified and the coefficients of the input system depend on parameters y1, łdots, yt, one also demonstrates that the corresponding one-block quantifier elimination problem can be solved in time (sn)O(dt).