Three flavors of extremal Betti tables

We discuss extremal Betti tables of resolutions in three different contexts. We begin over the graded polynomial ring, where extremal Betti tables correspond to pure resolutions. We then contrast this behavior with that of extremal Betti tables over regular local rings and over a bigraded ring.Comment: 20 page

Open Wilson Lines and Group Theory of Noncommutative Yang-Mills Theory in Two Dimensions

The correlation functions of open Wilson line operators in two-dimensional Yang-Mills theory on the noncommutative torus are computed exactly. The correlators are expressed in two equivalent forms. An instanton expansion involves only topological numbers of Heisenberg modules and enables extraction of the weak-coupling limit of the gauge theory. A dual algebraic expansion involves only group theoretic quantities, winding numbers and translational zero modes, and enables analysis of the strong-coupling limit of the gauge theory and the high-momentum behaviour of open Wilson lines. The dual expressions can be interpreted physically as exact sums over contributions from virtual electric dipole quanta.Comment: 37 pages. References adde

Borel Sets and Sectional Matrices

Following the path trodden by several authors along the border between Algebraic Geometry and Algebraic Combinatorics, we present some new results on the combinatorial struc- ture of Borel ideals. This enables us to prove theorems on the shape of the sectional matrix of a homogeneous ideal, which is a new invariant stronger than the Hilbert function

On the computation of Hilbert-Poincare` Series

We prove a theorem, which provides a formula for the computation of the Poincar\'e series of a monomial ideal in $k[X_1,\dots,X_n]$, via the computation of the Poincare' series of some monomial ideals in $k[X_1,\dots ,\widehat{X_i}, \dots,X_n]$. The complexity of our algorithm is optimal for Borel-normed ideals and an implementation in CoCoA strongly confirms its efficiency. An easy extension computes the Poincare' series of graded modules over standard algebras

Geometric Consequences of Extremal Behavior in a Theorem of Macaulay

F. S. Macaulay gave necessary and sufficient conditions on the growth of a nonnegative integer-valued function which determine when such a function can be the Hilbert function of a standard graded $k$-algebra. We investigate some algebraic and geometric consequences which arise from the extremal cases of Macaulay's theorem. Our work also builds on the fundamental work of G. Gotzmann. Our principal applications are to the study of Hilbert functions of zero-schemes with uniformity conditions. As a consequence, we have new strong limitations on the possible Hilbert functions of the points which arise as a general hyperplane section of an irreducible curve

Decompositions of binomial ideals

Algebraic statistics, Binomial ideals, Commuting birth and death ideals, Computational commutative algebra, Primary decomposition,