Bounds For Arithmetic Degrees

Let I be a monomial ideal of the polynomial ring R=k[x1,x2,···,xn], I∨be theAlexander duality of its polarization. We discuss the arithmetic degree of I and prove thefollowing interesting formula: adeg(R/I)=μ(I∨), whereμ(I∨) denotes the number of min-imal generators of I∨. By this formula, we obtain some new bounds for the degree and thearithmetic degree of a monomial ideal: adeg(I)≤deg(m1)· deg(m2)··· deg(mmht(I)). Thisconsequence is much better than discussed before in most cases.