Dimension and depth dependent upper bounds in polynomial ideal theory

Abstract

We improve certain upper bounds for the degree of Gröbner bases and the Castelnuovo-Mumford regularity of polynomial ideals. For the degree of Gröbner bases, we exclusively work in deterministically verifiable and achievable generic positions of a combinatorial nature, namely either strongly stable position or quasi stable position. Furthermore, we exhibit new dimension and depth depending upper bounds for the Castelnuovo-Mumford regularity and the degrees of the elements of the reduced Gröbner basis (w.r.t. the degree reverse lexicographical ordering) of a homogeneous ideal in these positions. Finally, it is shown that similar upper bounds hold in positive characteristic.