Uniform bounds on projective dimension and Castelnuovo-Mumford regularity

Abstract: In this article we obtain uniform effective upper bounds for the projective dimension and the Castelnuovo-Mumford regularity of homogeneous ideals inside a standard graded polynomial ring $S$ over a field. Such bounds are independent of the number of variables of $S$, in the spirit of Stillman's conjecture and of the Ananyan-Hochster's theorem, and depend on partial data extracted from the beginning or the end of the resolution. In this direction, we extend a result of McCullough from 2012 regarding a bound on regularity in terms of half the syzygies to a bound on the projective dimension and the regularity of an ideal in terms of a fraction of the syzygies.