2020/03/01 by Tigran Ananyan, Melvin Hochster, Ananyan, Tigran +1
Computer Science · Mathematics · #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2003.00589
openalex publication_date 2020/03/01 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
Let Rh denote the polynomial ring in variables x1, ,\…, , xh over\na specified field K. We consider all of these rings simultaneously, and in\neach use lexicographic (lex) monomial order with x1 > \⋯ > xh. Given a\nfixed homogeneous ideal I in Rh, for each d there is unique lex ideal\ngenerated in degree at most d whose Hilbert function agrees with the Hilbert\nfunction of I up to degree d. When we consider IRN for N \≥ h, the\nset mathfrakBd(I,N) of minimal generators for this lex ideal in degree at\nmost d may change, but mathfrakBd(I,N) is constant for all N \≫ 0.\nWe let mathfrakBd(I) denote the set of generators one obtains for all N\n\≫ 0, and we let bd = bd(I) be its cardinality. The sequences b1, ,\n\…, , bd, , \… obtained in this way may grow very fast. Remarkably,\neven when I = (x12, x22), one obtains a very interesting sequence, 0, 2,\n3, 4, 6, 12, 924, 409620, ,\…. This sequence is the same as Hd-1 + 1\nfor d \≥ 2, where Hd is the d ,th Hamilton number. The Hamilton\nnumbers were studied by Hamilton and by Hammond and Sylvester because of their\noccurrence in a counting problem connected with the use of Tschirnhaus\ntransformations in manipulating polynomial equations.\n