For the almost complete intersection ideals (x21, ... , x2n, (x1 + & centerdot; & centerdot; & centerdot; + xn)k), we compute their reduced Gr & ouml;bner basis for any term ordering, revealing a combinatorial structure linked to lattice paths, elementary symmetric polynomials, and Catalan numbers. Using this structure, we classify the weak Lefschetz property for these ideals. Additionally, we provide a new proof of the well-known result that the squarefree algebra satisfies the strong Lefschetz property. Finally, we compute the Betti numbers of the initial ideals and construct a minimal free resolution using a Mayer-Vietoris tree approach. 

We classify the minimal number of generators of artinian equigenerated monomial ideals I such that k[x1,…, xn]/I is forced to have the weak Lefschetz property. © 2024. American Mathematical Society. All rights reserved.

We consider homogeneous binomial ideals I=(f1,…,fn)𝐼=(𝑓1,…,𝑓𝑛) in K[x1,…,xn]𝐾[𝑥1,…,𝑥𝑛], where fi=aixdii−bimi𝑓𝑖=𝑎𝑖𝑥𝑖𝑑𝑖−𝑏𝑖𝑚𝑖 and ai≠0𝑎𝑖≠0. When such an ideal is a complete intersection, we show that the monomials which are not divisible by xdii𝑥𝑖𝑑𝑖 for i=1,…,n𝑖=1,…,𝑛 form a vector space basis for the corresponding quotient, and we describe the Macaulay dual generator in terms of a directed graph that we associate to I. These two properties can be seen as a natural generalization of well-known properties for monomial complete intersections. Moreover, we give a description of the radical of the resultant of I in terms of the directed graph.

We use Macaulay’s inverse system to study the Hilbert series for almost complete intersections generated by uniform powers of general linear forms. This allows us to give a classification of the weak Lefschetz property for these algebras, settling a conjecture by Migliore, Miró-Roig, and Nagel.

Computer Algebra relies heavily on the computation of Gröbner bases, and these computations are primarily performed by means of Buchberger’s algorithm. In this overview paper, we focus on methods avoiding the computational intensity associated to Buchberger’s algorithm and, in most cases, even avoiding the concept of Gröbner bases, in favour of methods relying on linear algebra and combinatorics.

We give a sharp lower bound for the Hilbert function in degree d of artinian quotients k[x₁,…,x_n]/I failing the Strong Lefschetz property, where I is a monomial ideal generated in degree d≥2. We also provide sharp lower bounds for other classes of ideals, and connect our result to the classification of the Hilbert functions forcing the Strong Lefschetz property by Zanello and Zylinski.

We introduce a general technique for decomposing monomial algebras which we use to study the Lefschetz properties. In particular, we prove that Gorenstein codimension three algebras arising from numerical semigroups have the strong Lefschetz property, and we give partial results on monomial almost complete intersections. We also study the reverse of the decomposition process – a gluing operation – which gives a way to construct monomial algebras with the Lefschetz properties.

In what follows, we pose two general conjectures about decompositions of homogeneous polynomials as sums of powers. The first one (suggested by G. Ottaviani) deals with the generic k-rank of complex-valued forms of any degree divisible by k in any number of variables. The second one (by the fourth author) deals with the maximal k-rank of binary forms. We settle the first conjecture in the cases of two variables and the second in the first non-trivial case of the 3-rd powers of quadratic binary forms.

We give a lower bound on the Hilbert series of the exterior algebra modulo a principal ideal generated by a generic form of odd degree and disprove a conjecture by Moreno-Socias and Snellman. We also show that the lower bound is equal to the minimal Hilbert series in some specific cases.

Based on the structure theory of pairs of skew-symmetric matrices, we give a conjecture for the Hilbert series of the exterior algebra modulo the ideal generated by two generic quadratic forms. We show that the conjectured series is an upper bound in the coefficient-wise sense, and we determine a majority of the coefficients. We also conjecture that the series is equal to the series of the squarefree polynomial ring modulo the ideal generated by the squares of two generic linear forms.