We show that a sufficiently general hypersurface of degree d in $\mathbb {P}^n$ admits a toric Gröbner degeneration after linear change of coordinates if and only if $d\leq 2n-1$ .

Spaces of power series solutions $y(\mathrm {t})$ in one variable $\mathrm {t}$ of systems of polynomial, algebraic, analytic or formal equations $f(\mathrm {t},\mathrm {y})=0$ can be viewed as ‘infinite-dimensional’ varieties over the ground field $\mathbf {k}$ as well as ‘finite-dimensional’ schemes over the power series ring $\mathbf {k}[[\mathrm {t}]]$ . We propose to call these solution spaces arquile varieties, as an enhancement of the concept of arc spaces. It will be proven that arquile varieties admit a natural stratification ${\mathcal Y}=\bigsqcup {\mathcal Y}_d$ , $d\in {\mathbb N}$ , such that each stratum ${\mathcal Y}_d$ is isomorphic to a Cartesian product ${\mathcal Z}_d\times \mathbb A^{\infty }_{\mathbf {k}}$ of a finite-dimensional, possibly singular variety ${\mathcal Z}_d$ over $\mathbf {k}$ with an affine space $\mathbb A^{\infty }_{\mathbf {k}}$ of infinite dimension. This shows that the singularities of the solution space of $f(\mathrm {t},\mathrm {y})=0$ are confined, up to the stratification, to the finite-dimensional part.

Our results are established simultaneously for algebraic, convergent and formal power series, as well as convergent power series with prescribed radius of convergence. The key technical tool is a linearisation theorem, already used implicitly by Greenberg and Artin, showing that analytic maps between power series spaces can be essentially linearised by automorphisms of the source space.

Instead of stratifying arquile varieties, one may alternatively consider formal neighbourhoods of their regular points and reprove with similar methods the Grinberg–Kazhdan–Drinfeld factorisation theorem for arc spaces in the classical setting and in the more general setting.

Geometric vertex decomposition and liaison are two frameworks that have been used to produce similar results about similar families of algebraic varieties. In this paper, we establish an explicit connection between these approaches. In particular, we show that each geometrically vertex decomposable ideal is linked by a sequence of elementary G-biliaisons of height $1$ to an ideal of indeterminates and, conversely, that every G-biliaison of a certain type gives rise to a geometric vertex decomposition. As a consequence, we can immediately conclude that several well-known families of ideals are glicci, including Schubert determinantal ideals, defining ideals of varieties of complexes and defining ideals of graded lower bound cluster algebras.

We study solutions of difference equations in the rings of sequences and, more generally, solutions of equations with a monoid action in the ring of sequences indexed by the monoid. This framework includes, for example, difference equations on grids (for example, standard difference schemes) and difference equations in functions on words. On the universality side, we prove a version of strong Nullstellensatz for such difference equations under the assumption that the cardinality of the ground field is greater than the cardinality of the monoid and construct an example showing that this assumption cannot be omitted. On the undecidability side, we show that the following problems are undecidable:

In this short note, we confirm a conjecture of Vasconcelos which states that the Rees algebra of any Artinian almost complete intersection monomial ideal is almost Cohen–Macaulay.

We investigate arithmetic, geometric and combinatorial properties of symmetric edge polytopes. We give a complete combinatorial description of their facets. By combining Gröbner basis techniques, half-open decompositions and methods for interlacing polynomials we provide an explicit formula for the $h^{\ast }$-polynomial in case of complete bipartite graphs. In particular, we show that the $h^{\ast }$-polynomial is $\unicode[STIX]{x1D6FE}$-positive and real-rooted. This proves Gal’s conjecture for arbitrary flag unimodular triangulations in this case, and, beyond that, we prove a strengthening due to Nevo and Petersen [On $\unicode[STIX]{x1D6FE}$-vectors satisfying the Kruskal–Katona inequalities. Discrete Comput. Geom.45(3) (2011), 503–521].

We compute Betti numbers for a Cohen–Macaulay tangent cone of a monomial curve in the affine $4$-space corresponding to a pseudo-symmetric numerical semigroup. As a byproduct, we also show that for these semigroups, being of homogeneous type and homogeneous are equivalent properties.

We study the decompositions of Hilbert schemes induced by the Schubert cell decomposition of the Grassmannian variety and show that Hilbert schemes admit a stratification into locally closed subschemes along which the generic initial ideals remain the same. We give two applications. First, we give completely geometric proofs of the existence of the generic initial ideals and of their Borel fixed properties. Second, we prove that when a Hilbert scheme of non-constant Hilbert polynomial is embedded by the Grothendieck–Plücker embedding of a high enough degree, it must be degenerate.

We upper-bound the number of common zeros over a finite grid of multivariate polynomials and an arbitrary finite collection of their consecutive Hasse derivatives (in a coordinate-wise sense). To that end, we make use of the tool from Gröbner basis theory known as footprint. Then we establish and prove extensions in this context of a family of well-known results in algebra and combinatorics. These include Alon's combinatorial Nullstellensatz [1], existence and uniqueness of Hermite interpolating polynomials over a grid, estimations of the parameters of evaluation codes with consecutive derivatives [20], and bounds on the number of zeros of a polynomial by DeMillo and Lipton [8], Schwartz [25], Zippel [26, 27] and Alon and Füredi [2]. As an alternative, we also extend the Schwartz-Zippel bound to weighted multiplicities and discuss its connection to our extension of the footprint bound.

We study an operation, that we call lifting, creating nonisomorphic monomial curves from a single monomial curve. Our main result says that all but finitely many liftings of a monomial curve have Cohen–Macaulay tangent cones even if the tangent cone of the original curve is not Cohen–Macaulay. This implies that the Betti sequence of the tangent cone is eventually constant under this operation. Moreover, all liftings have Cohen–Macaulay tangent cones when the original monomial curve has a Cohen–Macaulay tangent cone. In this case, all the Betti sequences are just the Betti sequence of the original curve.

This work generalises the short resolution given by Pisón Casares [‘The short resolution of a lattice ideal’, Proc. Amer. Math. Soc.131(4) (2003), 1081–1091] to any affine semigroup. We give a characterisation of Apéry sets which provides a simple way to compute Apéry sets of affine semigroups and Frobenius numbers of numerical semigroups. We also exhibit a new characterisation of the Cohen–Macaulay property for simplicial affine semigroups.

To a pair $P$ and $Q$ of finite posets we attach the toric ring $K[P,Q]$ whose generators are in bijection to the isotone maps from $P$ to $Q$. This class of algebras, called isotonian, are natural generalizations of the so-called Hibi rings. We determine the Krull dimension of these algebras and for particular classes of posets $P$ and $Q$ we show that $K[P,Q]$ is normal and that their defining ideal admits a quadratic Gröbner basis.

Let $I$ and $J$ be homogeneous ideals in a standard graded polynomial ring. We study upper bounds of the Hilbert function of the intersection of $I$ and $g(J)$, where $g$ is a general change of coordinates. Our main result gives a generalization of Green’s hyperplane section theorem.

Building on coprincipal mesoprimary decomposition [Kahle and Miller, Decompositions of commutative monoid congruences and binomial ideals, Algebra and Number Theory 8 (2014), 1297–1364], we combinatorially construct an irreducible decomposition of any given binomial ideal. In a parallel manner, for congruences in commutative monoids we construct decompositions that are direct combinatorial analogues of binomial irreducible decompositions, and for binomial ideals we construct decompositions into ideals that are as irreducible as possible while remaining binomial. We provide an example of a binomial ideal that is not an intersection of irreducible binomial ideals, thus answering a question of Eisenbud and Sturmfels [Binomial ideals, Duke Math. J. 84 (1996), 1–45].

We consider the classical problem of finding the best uniform approximation by polynomials of $1/(x-a)^2,$ where $a>1$ is given, on the interval $[-\! 1,1]$. First, using symbolic computation tools we derive the explicit expressions of the polynomials of best approximation of low degrees and then give a parametric solution of the problem in terms of elliptic functions. Symbolic computation is invoked then once more to derive a recurrence relation for the coefficients of the polynomials of best uniform approximation based on a Pell-type equation satisfied by the solutions.

Groebner bases for the ideals determining mod $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$ cohomology of the real flag manifolds $F(1,1,n)$ and $F(1,2,n)$ are obtained. These are used to compute appropriate Stiefel–Whitney classes in order to establish some new nonembedding and nonimmersion results for the manifolds $F(1,2,n)$.

Let $S$ be a polynomial ring over a field $K$ and let $I$ be a monomial ideal of $S$. We say that $I$ is MHC (that is, $I$ satisfies the maximal height condition for the associated primes of $I$) if there exists a prime ideal $\mathfrak{p}\in {\mathrm{Ass} }_{S} \hspace{0.167em} S/ I$ for which $\mathrm{ht} (\mathfrak{p})$ equals the number of indeterminates that appear in the minimal set of monomials generating $I$. Let $I= { \mathop{\bigcap }\nolimits}_{i= 1}^{k} {Q}_{i} $ be the irreducible decomposition of $I$ and let $m(I)= \max \{ \vert Q_{i}\vert - \mathrm{ht} ({Q}_{i} ): 1\leq i\leq k\} $, where $\vert {Q}_{i} \vert $ denotes the total degree of ${Q}_{i} $. Then it can be seen that when $I$ is primary, $\mathrm{reg} (S/ I)= m(I)$. In this paper we improve this result and show that whenever $I$ is MHC, then $\mathrm{reg} (S/ I)= m(I)$ provided $\vert {\mathrm{Ass} }_{S} \hspace{0.167em} S/ I\vert \leq 2$. We also prove that $m({I}^{n} )\leq n\max \{ \vert Q_{i}\vert : 1\leq i\leq ~k\} - \mathrm{ht} (I)$, for all $n\geq 1$. In addition we show that if $I$ is MHC and $w$ is an indeterminate which is not in the monomials generating $I$, then $\mathrm{reg} (S/ \mathop{(I+ {w}^{d} S)}\nolimits ^{n} )\leq \mathrm{reg} (S/ I)+ nd- 1$ for all $n\geq 1$ and $d$ large enough. Finally, we implement an algorithm for the computation of $m(I)$.

In 1992, Göran Björck and Ralf Fröberg completely characterized the solution set of cyclic-8. In 2001, Jean-Charles Faugère determined the solution set of cyclic-9, by computer algebra methods and Gröbner basis computation. In this paper, a new theory in matrix analysis of rank-deficient matrices together with algorithms in numerical algebraic geometry enables us to present a symbolic-numerical algorithm to derive exactly the defining polynomials of all prime ideals of positive dimension in primary decomposition of cyclic-12. Empirical evidence together with rigorous proof establishes the fact that the positive-dimensional solution variety of cyclic-12 just consists of 72 quadrics of dimension one.

Two fundamental questions in the theory of Gröbner bases are decision (‘Is a basis G of a polynomial ideal a Gröbner basis?’) and transformation (‘If it is not, how do we transform it into a Gröbner basis?’) This paper considers the first question. It is well known that G is a Gröbner basis if and only if a certain set of polynomials (the S-polynomials) satisfy a certain property. In general there are m(m−1)/2 of these, where m is the number of polynomials in G, but criteria due to Buchberger and others often allow one to consider a smaller number. This paper presents two original results. The first is a new characterization theorem for Gröbner bases that makes use of a new criterion that extends Buchberger’s criteria. The second is the identification of a class of polynomial systems G for which the new criterion has dramatic impact, reducing the worst-case scenario from m(m−1)/2 S-polynomials to m−1.

Let f be a probability density function on (a, b) ⊂ (0, ∞), and consider the class Cf of all probability density functions of the form Pf, where P is a polynomial. Assume that if X has its density in Cf then the equilibrium probability density x ↦ P(X > x) / E(X) also belongs to Cf: this happens, for instance, when f(x) = Ce−λx or f(x) = C(b − x)λ−1. We show in the present paper that these two cases are the only possibilities. This surprising result is achieved with an unusual tool in renewal theory, by using ideals of polynomials.