We study multiplicities of jumping numbers multiplier ideals in a smooth variety arbitrary dimension. prove that the multiplicity function is quasi-polynomial, hence proving Poincar\'e series rational function. further when various components quasi-polynomial have highest possible degree and relate it to contributed by Rees valuations. Finally, we special case monomial ideals.

In this paper, by a modification of a previously constructed minimal free resolution for a transversal monomial ideal, the Betti numbers of this ideal is explicitly computed. For convenient characteristics of the ground field, up to a change of coordinates, the ideal of t-minors of a generic pluri-circulant matrix is a transversal monomial ideal . Using a Gröbner basis for this ideal, it is sho...

When do syzygies depend on the characteristic of field? Even for well-studied families examples, very little is known. For a family random monomial ideals, namely Stanley--Reisner ideals flag complexes, we prove that Betti numbers asymptotically almost always characteristic. Using this result, also develop heuristic dependence asymptotic algebraic varieties.