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Eisenbud. Commutative algebra with a view toward algebraic geometry. Springer, 1995. [7] M. Giusti. Some effectivity problems in polynomial ideal theory. In Eurosam, volume 84, pages 159–171. Springer, 1984. -M. Greuel, G. Pfister, and H. Sch¨ onemann. 0 — A computer algebra system for polynomial computations. 2009. de. [9] M. Kratzer. Computing the dimension of a polynomial ideal and membership in low-dimensional ideals. Bachelor’s thesis, Technische Universit¨ at M¨ unchen, 2008. [10] M. Kreuzer and L.
C∈P with Jk = g1 , . . , gn−r , f1 , . . , fk−1 . i=1 for k = 0, . . , s − 1. The equality I = Js then yields the stated result. The ”⊃”-inclusion is clear since fj , gj ∈ I. For the other inclusion, the case k = 0 is trivial. So assume k > 0. Let f ∈ Jk+1 and thus f = f + a · fk = (f + a · (fk − nf J (fk ))) + a · nf J (fk ) with f , a · (fk − nf J (fk )) ∈ Jk . We rewrite #{C ∈ P : dim(C) = 0, deg(C) = z} = ϕP (z) − ϕP (z) = (ϕK[X] (z) − ϕJ (z)) − (ϕK[X] (z) − ϕJ (z)) = 0 Thus there are no cones with degree greater or equal max{a1 , d1 + .
Computational commutative algebra 2. 2005. [11] K. K¨ uhnle and E. Mayr. Exponential space computation of Gr¨ obner bases. In Proceedings of the 1996 international symposium on Symbolic and algebraic computation, pages 63–71. ACM New York, NY, USA, 1996. [12] D. Lazard. Gr¨ obner bases, Gaussian elimination and resolution of systems of algebraic equations. In Proc. EUROCAL, volume 83, pages 146–156. Springer, 1983. [13] E. Mayr and A. Meyer. The complexity of the word problems for commutative semigroups and polynomial ideals.