Guillermo Matera and Jose Maria Turull Torres. The Space Complexity of Elimination Theory: Upper Bounds. In F. Cucker and M. Shub, editors, Foundations of Computational Mathematics (Selected Papers of a Conference held at IMPA in Rio de Janeiro), pages 267--276, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the Dimension of a Polynomial Ideal Anna Bernasconi, Ernst W. Mayr, Michal Mnuk and Martin Raab Institut fur Informatik Technische Universitat Munchen 80290 Munich, Germany January 9, 2002 Abstract Following ideas from [Hei83, DFGS91, MT97] and applying the techniques proposed in [May89, KM96, Kuh98] we present a deterministic algorithm for computing the dimension of a polynomial ideal requiring polynomial working space. 1 Introduction The problem of computing the dimension of an ideal has been investigated in a number of ....

....proposed in [May89, KM96, Kuh98] we present a deterministic algorithm for computing the dimension of a polynomial ideal requiring polynomial working space. 1 Introduction The problem of computing the dimension of an ideal has been investigated in a number of papers, see for instance [KW88, DFGS91, MT97, Koi97, Kuh98]. In particular, in [MT97] a deterministic algorithm for computing the dimension of a polynomial ideal requiring polynomial working space is presented. This algorithm is based on ideas from [DFGS91] which in turn relies on results from [Hei83] In this paper we review these results with the aim of ....

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Guillermo Matera and Jose Maria Turull Torres. The Space Complexity of Elimination Theory: Upper Bounds. In F. Cucker and M. Shub, editors, Foundations of Computational Mathematics (Selected Papers of a Conference held at IMPA in Rio de Janeiro), pages 267--276, 1997.

....sets. In the complex case it is known that the dimension can always be computed within that time bound (and in fact in time s O(1) D O(n) For instance this follows from the fact that the randomized reduction in [13] produces existential formulas with only O(n) variables (see also [8, 10, 11, 15]) It is by no means clear whether a similarly parsimonious reduction exists in the real case. If this question turns out to have a positive answer, a (sD) O(n) bound for DIMR can be expected. On the other hand, as we have already pointed out in section 3.1, life is sometimes easier over the ....

G. Matera and J. Torres. The space complexity of elimination theory: Upper bounds. In F. Cucker and M. Shub, editors, Foundations of Computational Mathematics (Selected Papers of a Conference Held at IMPA in Rio de Janeiro), pages 267--276, 1997.

....case. More precisely, the Hauptproblem der Idealtheorie turns out to be complete in exponential (memory) space (see [33, 43, 8] On the other hand, almost all of the most fundamental problems of computational classical algebraic geometry are proved to be solvable in polynomial space (see e.g. [32, 9, 30, 29, 25]) Thus computational complexity is able to distinguish between geometry and algebra and supports the viewpoint of Kronecker (geometry) against the viewpoint of Hilbert and Macaulay (algebra) It is well known that Kronecker s personality was highly con ictive for his time. It is less known how ....

Guillermo Matera and Jose Maria Turull Torres. The space complexity of elimination theory: upper bounds. In Foundations of computational mathematics (Rio de Janeiro, 1997), pages 267-276. Springer, Berlin, 1997.

....sets. In the complex case it is known that the dimension can always be computed within that time bound (and in fact in time s O(1) D O(n) For instance this follows from the fact that the randomized reduction in [14] produces existential formulas with only O(n) variables (see also [8, 10, 11, 15]) It is by no means clear whether a similarly parsimonious reduction exists in the real case. If this question turns out to have a positive answer, a (sD) O(n) bound for DIMR can be expected. On the other hand, as we have already pointed out in section 3.1, life is sometimes easier over the ....

G. Matera and J. Torres. The space complexity of elimination theory: Upper bounds. In F. Cucker and M. Shub, editors, Foundations of Computational Mathematics (Selected Papers of a Conference Held at IMPA in Rio de Janeiro), pages 267--276, 1997.

....from the randomized reduction that DIM 2 AM as well. Without GRH it is only known that HN 2 PSPACE, and that HNC can be solved in parallel polynomial time. It follows from this randomized reduction that the same bounds apply to DIM and DIMC . This is not a new result: see [7, 11, 12] and also [19] where special attention is paid to uniformity issues. Another early reference for that problem (with less emphasis on complexity issues) is [18] The PSPACE algorithm described in section 4.3 seems to be simpler than previously published algorithms. In [8] and [12] polynomial time means ....

....then convert it into a space efficient algorithm using the equivalence between parallel time and sequential space discovered by Borodin [5] This strategy could be carried out with our present algorithm. We will not go into the details since they can be found in many recent papers, for instance [19]. That paper gives a O(n 4 log 2 (LsD) space bound for both HN and DIM (and also for Noether Normalization) For DIMC , Theorem 4.2 would translate into a polynomial depth bound for uniform arithmetic circuits. In terms of sequential complexity, our enumeration procedure implies a uniform ....

G. Matera and J. Torres. The space complexity of elimination theory: Upper bounds. In F. Cucker and M. Shub, editors, Foundations of Computational Mathematics (Selected Papers of a Conference Held at IMPA in Rio de Janeiro), pages 267--276, 1997.

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