It is well known that a monomial complete intersection has the strong Lefschetz property in characteristic zero. This property is equivalent to the statement that any power of the sum of the variables is a maximal rank element on the complete intersection. In this paper, we investigate what happens when this element is replaced by another symmetric polynomial, in an equigenerated complete intersection.

We answer the question completely for the power sum symmetric polynomial using a grading technique, and for any Schur polynomial in the case of two variables by deriving a closed formula for the determinants of a family of Toeplitz matrices. Further, we obtain partial results in three or more variables for the elementary and the complete homogeneous symmetric polynomials and pose several open questions.

Motivated by the foundational result that a monomial complete intersection has the strong Lefschetz property (SLP) in characteristic zero, it is natural to ask when monomial almost complete intersections have the SLP. In this paper, using the Hilbert series as a central tool, we investigate the strong Lefschetz property for certain monomial almost complete intersections, those with the non-pure-power generator having support in two variables, and those with symmetric Hilbert series. In the former case, we give a complete classification for when the SLP holds, and in the latter case, we prove that such algebras always have the SLP.

We study almost complete intersection ideals in a polynomial ring, generated by powers of all the variables together with a power of their sum. Our main result is an explicit description of the reduced Gröbner bases for these ideals under any term order. Our approach is primarily combinatorial, focusing on the structure of the initial ideal. We associate a lattice path to each monomial in the vector space basis of an Artinian monomial complete intersection and introduce a reflection operation on these paths, which enables a key counting argument. As a consequence, we provide a new proof that Artinian monomial complete intersections possess the strong Lefschetz property over fields of characteristic zero. Our results also offer new insights into the longstanding problem of classifying the weak Lefschetz property for such intersections in characteristic p. Furthermore, we show that the number of Gröbner basis elements in each degree is connected to several well-known sequences, including the (generalized) Catalan, Motzkin, and Riordan numbers, and connect these numbers to the study of entanglement detection in spin systems within quantum physics. 

For the almost complete intersection ideals (x_1^2, ..., x_n^2, (x_1+...+x_n)^k) we compute their reduced Gröbner 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.

For an ideal I in a Noetherian ring R, we introduce and study its conductor as a tool to explore the Rees algebra of I. The conductor of I is an ideal C(I) ⊂ R obtained from the defining ideals of the Rees algebra and the symmetric algebra of I by a colon operation. Using this concept we investigate when adding an element to an ideal preserves the property of being of linear type. In this regard, a generalization of a result by Valla in terms of the conductor ideal is presented. When the conductor of a graded ideal in a polynomial ring is the graded maximal ideal, a criteria is given for when the Rees algebra and the symmetric algebra have the same Krull dimension. Finally, noting the fact that the conductor of a monomial ideal is a monomial ideal, the conductor of some families of monomial ideals, namely bounded Veronese ideals and edge ideals of graphs, are determined. 

Consider a standard graded artinian k-algebra B and an extension of B by a new variable, A = B ⊗ k[x]/(x^d) for some d ≥ 1. We will show how maximal rank properties for powers of a general linear form on A can be determined by maximal rank properties for different powers of general linear forms on B. This is then used to study Lefschetz properties of algebras that can be obtained via such extensions. In particular, it allows for a new proof that monomial complete intersections have the strong Lefschetz property over a field of characteristic zero. Moreover, it gives a recursive formula for the determinants that show up in that case. Finally, for algebras over a field of characteristic zero, we give a classification for what properties B must have for all extensions B ⊗ k[x]/(x^d) to have the weak or the strong Lefschetz property. 

Consider ideals I of the form I = (x_1^2 , . . . , x_n^2 ) + RLex(x_ix_j ) where RLex(x_ix_j ) is the ideal generated by all the square-free monomials which are greater than or equal to x_ix_j in the reverse lexicographic order. We will determine some interesting properties regarding the shape of the Hilbert series of I. Using a theorem of Lindsey, this allows for a short proof that any algebra defined by I has the strong Lefschetz property when the underlying field is of characteristic zero. Building on recent work by Phuong and Tran, this result is then extended to fields of sufficiently high positive characteristic. As a consequence, this shows that for any possible number of minimal generators for an artinian quadratic ideal there exists such an ideal minimally generated by that many monomials and defining an algebra with the strong Lefschetz property. 

We consider homogeneous binomial ideals I = (f_1, . . . , f_n) in K[x_1, . . . , xn], where f_i = a_ix^{d_i} − b_im_i and a_i \neq 0. When such an ideal is a complete intersection, we show that the monomials which are not divisible by x_i^{d_i}  for i = 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.