Initially regular sequences and depths of ideals

For an arbitrary ideal I in a polynomial ring R we define the notion of initially regular sequences on R/I. These sequences share properties with regular sequences. In particular, the length of an initially regular sequence provides a lower bound for the depth of R/I when I is homogeneous. Using combinatorial information from the initial ideal of I we construct sequences of linear polynomials that form initially regular sequences on R/I. We identify situations where initially regular sequences are also regular sequences, and we show that our results can be combined with polarization to improve known depth bounds for general monomial ideals.