In computational algebra initial ideals and particularly generic initial ideals are monomial ideals used for estimating a host of algebraic and homological properties of graded ideals. In this talk we employ generic initial ideals to introduce new invariants which measure the complexity of homogeneous ideals. These invariants are termed axial constants because they represent the points in which the set of exponents of monomials in the generic initial ideal intersects the coordinate axes. They simultaneously generalize the initial degree and the Castelnuovo-Mumford regularity. We give homological and ideal-theoretic interpretations for this family of constants. This is joint work with the members of the PolymathJr group on computational algebra: DeBellevue, Lebovitz, Li, Lotfi, Mohite, Pan, Pathak, Roshan Zamir, and Zhang.

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