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Let $X$ be an affine scheme of $k \times \mathbb{N}$-matrices and $Y$ be an affine scheme of $\mathbb{N} \times \cdots \times \mathbb{N}$-dimensional tensors. The group Sym$(\mathbb{N})$ acts naturally on both $X$ and $Y$ and on their coordinate rings. We show that the Zariski closure of the image of a Sym$(\mathbb{N})$-equivariant morphism of schemes from $X$ to $Y$ is defined by finitely many Sym$(\mathbb{N})$-orbits in the coordinate ring of $Y$. Moreover, we prove that the closure of the image of this map is Sym$(\mathbb{N})$-Noetherian, that is, every descending chain of Sym$(\mathbb{N})$-stable closed subsets stabilizes.

We show that the shortest nonzero polynomials vanishing on bounded-rank matrices and skew-symmetric matrices are the determinants and Pfaffians characterising the rank. Algebraically, this means that in the ideal generated by all $t$-minors or $t$-Pfaffians of a generic matrix or skew-symmetric matrix one cannot find any polynomial with fewer terms than those determinants or Pfaffians, respectively, and that those determinants and Pfaffians are essentially the only polynomials in the ideal with that many terms. As a key tool of independent interest, we show that the ideal of a sufficiently general $t$-dimensional subspace of an affine $n$-space does not contain polynomials with fewer than $t+1$ terms., Comment: 13 pages, comments welcome, v2: 15 pages, final version as in Bulletin LMS

A theorem due to Kazhdan and Ziegler implies that, by substituting linear forms for its variables, a homogeneous polynomial of sufficiently high strength specialises to any given polynomial of the same degree in a bounded number of variables. Using entirely different techniques, we extend this theorem to arbitrary polynomial functors. As a corollary of our work, we show that specialisation induces a quasi-order on elements in polynomial functors, and that among the elements with a dense orbit there are unique smallest and largest equivalence classes in this quasi-order., Comment: 19 pages

We show that if X_n is a variety of cxn-matrices that is stable under the group Sym([n]) of column permutations and if forgetting the last column maps X_n into X_{n-1}, then the number of Sym([n])-orbits on irreducible components of X_n is a quasipolynomial in n for all sufficiently large n. To this end, we introduce the category of affine FI^op-schemes of width one, review existing literature on such schemes, and establish several new structural results about them. In particular, we show that under a shift and a localisation, any width-one FI^op-scheme becomes of product form, where X_n=Y^n for some scheme Y in affine c-space. Furthermore, to any FI^op-scheme of width one we associate a component functor from the category FI of finite sets with injections to the category PF of finite sets with partially defined maps. We present a combinatorial model for these functors and use this model to prove that Sym([n])-orbits of components of X_n, for all n, correspond bijectively to orbits of a groupoid acting on the integral points in certain rational polyhedral cones. Using the orbit-counting lemma for groupoids and theorems on quasipolynomiality of lattice point counts, this yields our Main Theorem., Comment: final version, 40 pages, added several examples and clarifications and corrected typos---with many thanks to a referee!

In a previous paper, the third author proved that finite-degree polynomial functors over infinite fields are topologically Noetherian. In this paper, we prove that the same holds for polynomial functors from free $R$-modules to finitely generated $R$-modules, for any commutative ring $R$ whose spectrum is Noetherian. As Erman-Sam-Snowden pointed out, when applying this with $R = \mathbb{Z}$ to direct sums of symmetric powers, one of their proofs of a conjecture by Stillman becomes characteristic-independent. Our paper advertises and further develops the beautiful but not so well-known machinery of polynomial laws. In particular, to any finitely generated R-module M we associate a topological space, which we show is Noetherian when $\operatorname{Spec}(R)$ is; this is the degree-zero case of our result on polynomial functors., Comment: 35 pages, 1 figure

Every tropical ideal in the sense of Maclagan-Rinc\'on has an associated tropical variety, a finite polyhedral complex equipped with positive integral weights on its maximal cells. This leads to the realisability question, ubiquitous in tropical geometry, of which weighted polyhedral complexes arise in this manner. Using work of Las Vergnas on the non-existence of tensor products of matroids, we prove that there is no tropical ideal whose variety is the Bergman fan of the direct sum of the V\'amos matroid and the uniform matroid of rank two on three elements, and in which all maximal cones have weight one., Comment: 11 pages; v2 includes two short new sections, main theorem is stated in slightly more generality

We introduce the monic rank of a vector relative to an affine-hyperplane section of an irreducible Zariski-closed affine cone $X$. We show that the monic rank is finite and greater than or equal to the usual $X$-rank. We describe an algorithmic technique based on classical invariant theory to determine, in concrete situations, the maximal monic rank. Using this technique, we establish three new instances of a conjecture due to B. Shapiro which states that a binary form of degree $d\cdot e$ is the sum of $d$ $d$-th powers of forms of degree $e$. Furthermore, in the case where $X$ is the cone of highest weight vectors in an irreducible representation---this includes the well-known cases of tensor rank and symmetric rank---we raise the question whether the maximal rank equals the maximal monic rank. We answer this question affirmatively in several instances., Comment: 26 pages, added a discussion on the monic rank for reducible cones

Using recent work by Erman-Sam-Snowden, we show that finitely generated ideals in the ring of bounded-degree formal power series in infinitely many variables have finitely generated Gr\"obner bases relative to the graded reverse lexicographic order. We then combine this result with the first author's work on topological Noetherianity of polynomial functors to give an algorithmic proof of the following statement: ideals in polynomial rings generated by a fixed number of homogeneous polynomials of fixed degrees only have a finite number of possible generic initial ideals, independently of the number of variables that they involve and independently of the characteristic of the ground field. Our algorithm outputs not only a finite list of possible generic initial ideals, but also finite descriptions of the corresponding strata in the space of coefficients., Comment: Several minor edits

We prove that any finite-degree polynomial functor is topologically Noetherian. This theorem is motivated by the recent resolution of Stillman's conjecture and a recent Noetherianity proof for the space of cubics. Via work by Erman-Sam-Snowden, our theorem implies Stillman's conjecture and indeed boundedness of a wider class of invariants of ideals in polynomial rings with a fixed number of generators of prescribed degrees., Comment: Final version

We prove that iterated toric fibre products from a finite collection of toric varieties are defined by binomials of uniformly bounded degree. This implies that Markov random fields built up from a finite collection of finite graphs have uniformly bounded Markov degree., Comment: several improvements, final version

The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimisation, complexity theory, and scientific computing. Motivated by recent developments in this last area, we introduce the notion of a uniform determinantal representation, not of a single polynomial but rather of all polynomials in a given number of variables and of a given maximal degree. We derive a lower bound on the size of the matrix, and present a construction achieving that lower bound up to a constant factor as the number of variables is fixed and the degree grows. This construction marks an improvement upon a recent construction due to Plestenjak-Hochstenbach, and we investigate the performance of new representations in their root-finding technique for bivariate systems. Furthermore, we relate uniform determinantal representations to vector spaces of singular matrices, and we conclude with a number of future research directions., Comment: 23 pages, 3 figures, 4 tables

We consider the degrees of the elements of a homogeneous system of parameters for the ring of invariants of a binary form, give a divisibility condition, and a complete classification for forms of degree at most 8., Comment: 15 pages, 1 table

Every Grassmannian, in its Pl\"ucker embedding, is defined by quadratic polynomials. We prove a vast, qualitative, generalisation of this fact to what we call Pl\"ucker varieties. A Pl\"ucker variety is in fact a family of varieties in exterior powers of vector spaces that, like the Grassmannian, is functorial in the vector space and behaves well under duals. A special case of our result says that for each fixed natural number k, the k-th secant variety of any Pl\"ucker-embedded Grassmannian is defined in bounded degree independent of the Grassmannian. Our approach is to take the limit of a Pl\"ucker variety in the dual of a highly symmetric space known as the infinite wedge, and to prove that up to symmetry the limit is defined by finitely many polynomial equations. For this we prove the auxilliary result that for every natural number p the space of p-tuples of infinite-by-infinite matrices is Noetherian modulo row and column operations. Our results have algorithmic counterparts: every bounded Pl\"ucker variety has a polynomial-time membership test, and the same holds for Zariski-closed, basis-independent properties of p-tuples of matrices., Comment: 25 pages, 4 figures

We consider a large class of monomial maps respecting an action of the infinite symmetric group, and prove that the toric ideals arising as their kernels are finitely generated up to symmetry. Our class includes many important examples where Noetherianity was recently proved or conjectured. In particular, our results imply Hillar-Sullivant's Independent Set Theorem and settle several finiteness conjectures due to Aschenbrenner, Martin del Campo, Hillar, and Sullivant. We introduce a matching monoid and show that its monoid ring is Noetherian up to symmetry. Our approach is then to factorize a more general equivariant monomial map into two parts going through this monoid. The kernels of both parts are finitely generated up to symmetry: recent work by Yamaguchi-Ogawa-Takemura on the (generalized) Birkhoff model provides an explicit degree bound for the kernel of the first part, while for the second part the finiteness follows from the Noetherianity of the matching monoid ring., Comment: 20 pages

Exploiting symmetry in Groebner basis computations is difficult when the symmetry takes the form of a group acting by automorphisms on monomials in finitely many variables. This is largely due to the fact that the group elements, being invertible, cannot preserve a term order. By contrast, inspired by work of Aschenbrenner and Hillar, we introduce the concept of equivariant Groebner basis in a setting where a_monoid_ acts by_homomorphisms_ on monomials in potentially infinitely many variables. We require that the action be compatible with a term order, and under some further assumptions derive a Buchberger-type algorithm for computing equivariant Groebner bases. Using this algorithm and the monoid of strictly increasing functions N -> N we prove that the kernel of the ring homomorphism R[y_{ij} | i,j in N, i > j] -> R[s_i,t_i | i in N], y_{ij} -> s_i s_j + t_i t_j is generated by two types of polynomials: off-diagonal 3x3-minors and pentads. This confirms a conjecture by Drton, Sturmfels, and Sullivant on the Gaussian two-factor model from algebraic statistics., Comment: 11 pages, improved exposition

This paper deals with two families of algebraic varieties arising from applications. First, the k-factor model in statistics, consisting of n-times-n covariance matrices of n observed Gaussian variables that are pairwise independent given k hidden Gaussian variables. Second, chirality varieties inspired by applications in chemistry. A point in such a chirality variety records chirality measurements of all k-subsets among an n-set of ligands. Both classes of varieties are given by a parameterisation, while for applications having polynomial equations would be desirable. For instance, such equations could be used to test whether a given point lies in the variety. We prove that in a precise sense, which is different for the two classes of varieties, these equations are finitely characterisable when k is fixed and n grows., Comment: 13 pages