Bounds for Regularity of Toric Varieties in Terms of Multiplicity

Abstract: The Eisenbud-Goto conjecture stated that for a homogeneous prime ideal in a polynomial ring over an algebraically closed field, the Castelnuovo-Mumford regularity is bounded above by the multiplicity minus height plus one. In 2017, McCullough and Peeva showed that the regularity of an arbitrary homogeneous prime cannot be bounded by any polynomial function of the multiplicity. For ideals generated by polynomials with at most $c$ terms, we present a bound on the regularity which is less than triple exponential in $c$ and the multiplicity. Furthermore, for toric ideals we get a bound that is smaller than double exponential in terms of multiplicity.