Computing bases of complete intersection rings in Noether position
Let k be an effective infinite perfect field, k[x1,...,xn] the polynomial ring in n variables and F∈k[x1,...,xn]M×M a square polynomial matrix verifying F2=F. Suppose that the entries of F are polynomials given by a straight-line program of size L and their total degrees are bounded by an integer D....
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| Autores principales: | , , , |
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| Formato: | JOUR |
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| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_00224049_v162_n2-3_p127_Almeida |
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| Sumario: | Let k be an effective infinite perfect field, k[x1,...,xn] the polynomial ring in n variables and F∈k[x1,...,xn]M×M a square polynomial matrix verifying F2=F. Suppose that the entries of F are polynomials given by a straight-line program of size L and their total degrees are bounded by an integer D. We show that there exists a well parallelizable algorithm which computes bases of the kernel and the image of F in time (nL)O(1)(MD)O(n). By means of this result we obtain a single exponential algorithm to compute a basis of a complete intersection ring in Noether position. More precisely, let f1,...,fn-r∈k[x1,...,xn] be a regular sequence of polynomials given by a slp of size ℓ, whose degrees are bounded by d. Let Rk[x1,...,xr] and Sk[x1,...,xn]/(f1,...,fn-r) such that S is integral over R; we show that there exists an algorithm running in time O(n)ℓdO(n2) which computes a basis of S over R. Also, as a consequence of our techniques, we show a single exponential well parallelizable algorithm which decides the freeness of a finite k[x1,...,xn]-module given by a presentation matrix, and in the affirmative case it computes a basis. © 2001 Elsevier Science B.V. |
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