Algebraic Complexity Theory (original) (raw)
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The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant pro-Typeset by the translator. Edited and reformatted by LE-TeX, Leipzig, using a Springer L A T E X macro package.
Real Algebraic Numbers: Complexity Analysis and Experimentation
Lecture Notes in Computer Science, 2008
We present algorithmic, complexity and implementation results concerning real root isolation of a polynomial of degree d, with integer coefficients of bit size ≤ τ , using Sturm (-Habicht) sequences and the Bernstein subdivision solver. In particular, we unify and simplify the analysis of both methods and we give an asymptotic complexity bound of e OB(d 4 τ 2 ). This matches the best known bounds for binary subdivision solvers. Moreover, we generalize this to cover the non square-free polynomials and show that within the same complexity we can also compute the multiplicities of the roots. We also consider algorithms for sign evaluation, comparison of real algebraic numbers and simultaneous inequalities (SI) and we improve the known bounds at least by a factor of d. Finally, we present our C++ implementation in synaps and some experimentations on various data sets.
[Victor V. Prasolov] Polynomials (Algorithms and C)
The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant pro-Typeset by the translator. Edited and reformatted by LE-TeX, Leipzig, using a Springer L A T E X macro package.
The Hardness of Polynomial Equation Solving
Foundations of Computational Mathematics, 2003
Elimination theory was at the origin of algebraic geometry in the nineteenth century and now deals with the 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. Our main result is the following: let there be given such a data structure and together with this data structure a universal elimination algorithm, say P, solving arbitrary parametric polynomial equation systems. Suppose that the algorithm P avoids “unnecessary” branchings and that P admits the efficient computation of certain natural limit objects (as, e.g., the Zariski closure of a given constructible algebraic set or the parametric greatest common divisor of two given algebraic families of univariate polynomials). Then P$ cannot be a polynomial time algorithm. The paper contains different variants of this result and discusses their practical implications.
On the k-error linear complexity of binary sequences derived from polynomial quotients
Science China Information Sciences, 2015
We investigate the k-error linear complexity of p 2-periodic binary sequences defined from the polynomial quotients (including the well-studied Fermat quotients), which is defined by q p,w (u) ≡ u w − u wp p mod p with 0 ≤ q p,w (u) ≤ p − 1, u ≥ 0, where p is an odd prime and 1 ≤ w < p. Indeed, first for all integers k, we determine exact values of the k-error linear complexity over the finite field F 2 for these binary sequences under the assumption of 2 being a primitive root modulo p 2 , and then we determine their k-error linear complexity over the finite field F p for either 0 ≤ k < p when w = 1 or 0 ≤ k < p − 1 when 2 ≤ w < p. Theoretical results obtained indicate that such sequences possess 'good' error linear complexity.
A Survey on the Complexity of Solving Algebraic Systems
International Mathematical Forum, 2010
This paper presents a lecture on existing algorithms for solving polynomial systems with their complexity analysis from our experiments on the subject. It is based on our studies of the complexity of solving parametric polynomial systems. It is intended to be useful to two groups of people: those who wish to know what work has been done and those who would like to do work in the field. It contains an extensive bibliography to assist readers in exploring the field in more depth. The paper provides different methods and techniques used for representing solutions of algebraic systems that include Rational Univariate Representations (RUR), Gröbner bases, etc.