The holonomic rank of the A-hypergeometric system HA(β) is shown to depend on the parameter vector β when the underlying toric ideal IA is a non Cohen Macaulay codimension 2 toric ideal. The set of exceptional parameters is usually infinite.

We prove that any rank two arithmetically CohenMacaulay vector bundle on a general hypersurface of degree at least six in P must be split.

The ideal of a Segre variety P1 × · · · × Pt ↪→ P(n1+1)···(nt+1)−1 is generated by the 2-minors of a generic hypermatrix of indeterminates (see [Ha1] and [Gr]). We extend this result to the case of SegreVeronese varieties. The main tool is the concept of “weak generic hypermatrix” which allows us to treat also the case of projection of Veronese surfaces from a set of general points and of Veron...

ii Contents 1 Lesson 1 : Macaulay versus Kronecker 3 1.1 Why does complex analysis interfere with algebraic problems ?. .. . 3 1.2 Some examples of classical division problems in K[X

We prove that the p-Quillen complex of a finite solvable group with cyclic derived group is Cohen-Macaulay, if p is an odd prime. If p = 2 we prove a similar conclusion, but there is a discussion to be made.

In 2007, Vallette built a bridge across posets and operads by proving that an operad is Koszul if only the associated partition are Cohen-Macaulay. Both notions of being Cohen–Macaulay admit different refinements: our goal here to link two these refinements. We more precisely prove any (basic-set) whose isomorphism-compatible CL-shellings admits Poincaré–Birkhoff–Witt basis. Furthermore,...

Abstract. We return to the algebraic roots of the problem of finding the solutions of a set of polynomial equations, and review this task from the linear algebra perspective. The system of polynomial equations is represented by a system of homogeneous linear equations by means of a structured Macaulay coefficient matrix multiplied by a vector containing monomials. Two properties are of key impo...