Bounds for Arithmetic Degrees |
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Authors: | Dancheng Lu Jun Yu |
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Institution: | 1. Department of Mathematics, Soochow University, Suzhou, P.R. Chinaludancheng@suda.edu.cn;3. Xishan High School, Wuxi, P.R. China |
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Abstract: | ABSTRACTLet I be a monomial ideal with minimal monomial generators m1,…, ms, and assume that deg(m1) ≥deg(m2) ≥ … ≥deg(ms). Among other things, we prove that the arithmetic degree of I is bounded above by deg(m1)…deg(mmht(I)), where mht(I) is the maximal height of associated primes of I. This bound is shaper than the one given in 12 Sturmfels, B., Trung, N. V., Vogel, W. (1995). Bounds on degrees of projective schemes. Math. Ann. 302:417–432.Crossref], Web of Science ®] , Google Scholar]] and more natural than the one given in 9 Hoa, L. T., Trung, N. V. (1998). On the Castelnuovo-Mumford regularity and the arithmetic degree of monomial ideals. Math. Z. 229:519–537.Crossref], Web of Science ®] , Google Scholar]]. In addition, we point out that adeg(I) ≠ adeg(Gin(I)) in general and conjecture that adeg(I) = adeg(Gin(I)) if and only if R/I is sequentially Cohen–Macaulay. |
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Keywords: | Alexander duality Arithmetic degree Minimal vertex cover Polarization |
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