A hyperplane restriction theorem and applications to reductions of ideals. (English) Zbl 1525.13038
Summary: Green’s general hyperplane restriction theorem gives a sharp upper bound for the Hilbert function of a standard graded algebra over an infinite field \(K\) modulo a general linear form. We strengthen Green’s result by showing that the linear forms that do not satisfy such estimate belong to a finite union of proper linear spaces. As an application we give a method to derive variations of the Eakin-Sathaye theorem on reductions. In particular, we recover and extend results by L. O’Carroll [J. Algebra 291, No. 1, 259–268 (2005; Zbl 1089.13012)] on the Eakin-Sathaye theorem for complete and joint reductions.
MSC:
| 13P05 | Polynomials, factorization in commutative rings |
| 13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |
| 13H15 | Multiplicity theory and related topics |
Citations:
Zbl 1089.13012References:
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