A symmetric ideal is an ideal in a polynomial ring which is stable under all permutations of the variables. In this paper we initiate a global study of zero-dimensional symmetric ideals. By this we mean a geometric study of the invariant Hilbert schemes Hilb(rho)(Sn)(C-n) parametrizing symmetric subschemes of C(n)whose coordinate rings, as S-n-modules, are isomorphic to a given representation rho. In the case that rho = ,M (lambda) is a permutation module corresponding to certain special types of partitions lambda of n, we prove that Hilb(rho)(Sn) (C-n) is irreducible or even smooth. We also prove irreducibility whenever dim rho <= 2nand the invariant Hilbert scheme is non-empty. In this same range, we classify all homogeneous symmetric ideals and decide which of these define singular points of Hilb(rho)(Sn) (C-n). A central tool is the combinatorial theory of higher Specht polynomials. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

The BKK theorem states that the mixed volume of the Newton polytopes of a system of polynomial equations upper bounds the number of isolated torus solutions of the system. Homotopy continuation solvers make use of this fact to pick efficient start systems. For systems where the mixed volume bound is not attained, such methods are still tracking more paths than necessary. We propose a strategy of improvement by lifting a system to an equivalent system with a strictly lower mixed volume at the expense of more variables. We illustrate this idea providing lifting constructions for arbitrary bivariate systems and certain dense-enough systems.

We study the ideal of submaximal minors of a sparse generic symmetric matrix. This ideal is generated by all (n−1)-minors of a symmetric n×n matrix whose entries in the upper triangle are distinct variables or zeros, and the zeros are only allowed at off-diagonal places. The surviving off-diagonal entries are encoded in a simple graph G with n vertices. We prove that the minimal free resolution of this ideal is obtained from the case without any zeros via a simple pruning procedure, extending methods of Boocher. This allows us to compute all graded Betti numbers in terms of n and a single invariant of G. Moreover, it turns out that these ideals are always radical and have Cohen–Macaulay quotients if and only if G is either connected or has no edges at all. The key input are some new Gröbner basis results with respect to non-diagonal term orders associated to G.

In this paper we study the novel notion of thin polytopes: lattice polytopes whose local $h^*$-polynomials vanish. The local $h^*$-polynomial is an important invariant in modern Ehrhart theory. Its definition goes back to Stanley with fundamental results achieved by Karu, Borisov & Mavlyutov, Schepers, and Katz & Stapledon. The study of thin simplices was originally proposed by Gelfand, Kapranov and Zelevinsky, where in this case the local $h^*$-polynomial simply equals its so-called box polynomial. Our main results are the complete classification of thin polytopes up to dimension 3 and the characterization of thinness for Gorenstein polytopes. The paper also includes an introduction to the local $h^*$-polynomial with a survey of previous results.

We study conditions on polynomials such that the ideal generated by their orbits under the symmetric group action becomes a monomial ideal or has a monomial radical. If the polynomials are homogeneous, we expect that such an ideal has a monomial radical if their coefficients are sufficiently general with respect to their supports. We prove this for instance in the case where some generator contains a power of a variable. Moreover, if the polynomials have only square-free terms and their coefficients do not sum to zero, then in a larger polynomial ring the ideal itself is square-free monomial. This has implications also for symmetric ideals of the infinite polynomial ring.

The set of smooth cubic hypersurfaces in ℙ^n is an open subset of a projective space. A compactification of the latter which allows to count the number of smooth cubic hypersurfaces tangent to a prescribed number of lines and passing through a given number of points is termed a 1–complete variety of cubic hypersurfaces, in analogy with the space of complete quadrics. Imitating the work of Aluffi for plane cubic curves, we construct such a space in arbitrary dimensions by a sequence of five blow-ups. The counting problem is then reduced to the computation of five total Chern classes. In the end, we derive the desired numbers in the case of cubic surfaces.

Gaussian double Markovian models consist of covariance matrices constrained by a pair of graphs specifying zeros simultaneously in the covariance matrix and its inverse. We study the semi-algebraic geometry of these models, in particular their dimension, smoothness and connectedness as well as algebraic and combinatorial properties.

We propose an explicit conjectural lift of the Neron–Severi Lie algebra of a hyperkähler variety X of $$K3^{[2]}$$ -type to the Chow ring of correspondences $$\mathrm{CH}^*(X \times X)$$ in terms of a canonical lift of the Beauville–Bogomolov class obtained by Markman. We give evidence for this conjecture in the case of the Hilbert scheme of two points of a K3 surface and in the case of the Fano variety of lines of a very general cubic fourfold. Moreover, we show that the Fourier decomposition of the Chow ring of X of Shen and Vial agrees with the eigenspace decomposition of a canonical lift of the cohomological grading operator.

We propose an explicit conjectural lift of the Neron-Severi Lie algebra of a hyperk\ahler variety $X$ of $K3^{[2]}$-type to the Chow ring of correspondences ${\rm CH}^\ast(X \times X)$ in terms of a canonical lift of the Beauville-Bogomolov class obtained by Markman. We give evidence for this conjecture in the case of the Hilbert scheme of two points of a $K3$ surface and in the case of the Fano variety of lines of a very general cubic fourfold. Moreover, we show that the Fourier decomposition of the Chow ring of $X$ of Shen and Vial agrees with the eigenspace decomposition of a canonical lift of the grading operator.