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SuperLinear TimeSpace Tradeoff Lower Bounds for Randomized Computation
, 2000
"... We prove the first timespace lower bound tradeoffs for randomized computation of decision problems. The bounds hold even in the case that the computation is allowed to have arbitrary probability of error on a small fraction of inputs. Our techniques are an extension of those used by Ajtai [Ajt99a, ..."
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Cited by 33 (0 self)
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We prove the first timespace lower bound tradeoffs for randomized computation of decision problems. The bounds hold even in the case that the computation is allowed to have arbitrary probability of error on a small fraction of inputs. Our techniques are an extension of those used by Ajtai [Ajt99a, Ajt99b] in his timespace tradeoffs for deterministic RAM algorithms computing element distinctness and for Boolean branching programs computing a natural quadratic form. Ajtai's bounds were of the following form...
The Hardness of Polynomial Equation Solving
, 2003
"... Elimination theory is at the origin of algebraic geometry in the 19th century and deals with algorithmic solving of multivariate polynomial equation systems over the complex numbers, or, more generally, over an arbitrary algebraically closed field. In this paper we investigate the intrinsic seq ..."
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Cited by 17 (9 self)
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Elimination theory is at the origin of algebraic geometry in the 19th century and deals with algorithmic solving of multivariate polynomial equation systems over the complex numbers, or, more generally, over an arbitrary algebraically closed field. In this paper we investigate the intrinsic sequential time complexity of universal elimination procedures for arbitrary continuous data structures encoding input and output objects of elimination theory (i.e. polynomial equation systems) and admitting the representation of certain limit objects.
Polynomial Equation Solving by Lifting Procedures for Ramified Fibers
, 2003
"... Let be given a parametric polynomial equation system which represents a generically {unrami ed family of zero{dimensional algebraic varieties. We exhibit an ecient algorithm which computes a complete description of the solution set of an arbitrary parameter instance from a complete description of t ..."
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Cited by 1 (1 self)
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Let be given a parametric polynomial equation system which represents a generically {unrami ed family of zero{dimensional algebraic varieties. We exhibit an ecient algorithm which computes a complete description of the solution set of an arbitrary parameter instance from a complete description of the in nitesimal structure of a particular rami ed parameter instance of our family. This generalizes in the case of space curves previous methods of Heintz et al. and Schost, which require the given parameter instance to be unrami ed. We illustrate our method solving particular polynomial equation systems by deformation techniques.
Lifting Procedures for Ramified Fibers and Polynomial Equation Solving
"... In [HKP+00] the following problem was solved: given a parametric polynomial equation system which represents a generically at and unramified family of zerodimensional algebraic varieties, and assuming that there is given a complete description of the solution of a particular unramified instance of ..."
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In [HKP+00] the following problem was solved: given a parametric polynomial equation system which represents a generically at and unramified family of zerodimensional algebraic varieties, and assuming that there is given a complete description of the solution of a particular unramified instance of the parametric system, produce efficiently a description of the parametric system. In this work we solve this problem under weaker hypotheses, namely admitting that the given zerodimensional variety can be ramified. In this case, assuming that a complete description of the infinitesimal structure of the ramified instance is given, we generalize the techniques used in [KT78] in order to solve the problem. We also analyze the complexity of the underlying algorithm and show a few examples where our techniques allow us to solve some polynomial equations systems in a very efficient way.