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40
On Fast Multiplication of Polynomials Over Arbitrary Algebras
 Acta Informatica
, 1991
"... this paper we generalize the wellknown SchonhageStrassen algorithm for multiplying large integers to an algorithm for multiplying polynomials with coefficients from an arbitrary, not necessarily commutative, not necessarily associative, algebra A. Our main result is an algorithm to multiply polyno ..."
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Cited by 151 (6 self)
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this paper we generalize the wellknown SchonhageStrassen algorithm for multiplying large integers to an algorithm for multiplying polynomials with coefficients from an arbitrary, not necessarily commutative, not necessarily associative, algebra A. Our main result is an algorithm to multiply polynomials of degree ! n in
When polynomial equation systems can be "solved" fast?
 IN PROC. 11TH INTERNATIONAL SYMPOSIUM APPLIED ALGEBRA, ALGEBRAIC ALGORITHMS AND ERRORCORRECTING CODES, AAECC11
, 1995
"... We present a new method for solving symbolically zerodimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of an alternative data structure: arithmetic networks and straightline programs with FOR gates. For sequential time complexity measu ..."
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Cited by 61 (19 self)
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We present a new method for solving symbolically zerodimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of an alternative data structure: arithmetic networks and straightline programs with FOR gates. For sequential time complexity measured by the size of these networks we obtain the following result: it is possible to solve any affine or toric zerodimensional equation system in nonuniform sequential time which is polynomial in the length of the input description and the "geometric degree " of the equation system. Here, the input is thought to be given by a straightline program (or alternatively in sparse representation), and the length of the input is measured by number of variables, degree of equations and size of the program (or sparsity of the equations). Geometric degree has to be adequately defined. It is always bounded by the algebraiccombinatoric "B'ezout number " of the system which is given by the Hilbert function of a suitable homogeneous ideal. However, in many important cases, the value of the geometric degree is much smaller than
Straightline programs in geometric elimination theory
 J. Pure Appl. Algebra
, 1998
"... Dedicated to Volker Strassen for his work on complexity We present a new method for solving symbolically zero–dimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of problem adapted data structures: arithmetic networks and straight–line prog ..."
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Cited by 57 (14 self)
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Dedicated to Volker Strassen for his work on complexity We present a new method for solving symbolically zero–dimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of problem adapted data structures: arithmetic networks and straight–line programs. For sequential time complexity measured by network size we obtain the following result: it is possible to solve any affine or toric zero–dimensional equation system in non–uniform sequential time which is polynomial in the length of the input description and the “geometric degree ” of the equation system. Here, the input is thought to be given by a straight–line program (or alternatively in sparse representation), and the length of the input is measured by number of variables, degree of equations and size of the program (or sparsity of the equations). The geometric degree of the input system has to be adequately defined. It is always bounded by the algebraic–combinatoric “Bézout number ” of the system which is given by the Hilbert function of a suitable homogeneous ideal. However, in many important cases, the value of the geometric
On The Complexity Of Computing Determinants
 COMPUTATIONAL COMPLEXITY
, 2001
"... We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n n matrix A with integer entries in (n and (n bi ..."
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Cited by 46 (17 self)
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We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n n matrix A with integer entries in (n and (n bit operations; here denotes the largest entry in absolute value and the exponent adjustment by "+o(1)" captures additional factors for positive real constants C 1 , C 2 , C 3 . The bit complexity (n results from using the classical cubic matrix multiplication algorithm. Our algorithms are randomized, and we can certify that the output is the determinant of A in a Las Vegas fashion. The second category of problems deals with the setting where the matrix A has elements from an abstract commutative ring, that is, when no divisions in the domain of entries are possible. We present algorithms that deterministically compute the determinant, characteristic polynomial and adjoint of A with n and O(n ) ring additions, subtractions and multiplications.
FOXBOX: A System for Manipulating Symbolic Objects in Black Box Representation
, 1998
"... The FOXBOX system puts in practice the black box representation of symbolic objects and provides algorithms for performing the symbolic calculus with such representations. Black box objects are stored as functions. For instance: a black box polynomial is a procedure that takes values for the variabl ..."
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Cited by 30 (12 self)
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The FOXBOX system puts in practice the black box representation of symbolic objects and provides algorithms for performing the symbolic calculus with such representations. Black box objects are stored as functions. For instance: a black box polynomial is a procedure that takes values for the variables as input and evaluates the polynomial at that given point. FOXBOX can compute the greatest common divisor and factorize polynomials in black box representation, producing as output new black boxes. It also can compute the standard sparse distributed representation of a black box polynomial, for example, one which was computed for an irreducible factor. We establish that the black box representation of objects can push the size of symbolic expressions far beyond what standard data structures could handle before. Furthermore, FOXBOX demonstrates the generic program design methodology. The FOXBOX system is written in C++. C++ template arguments provide for abstract domain types. Currently, F...
Factorization of Polynomials Given by StraightLine Programs
 Randomness and Computation
, 1989
"... An algorithm is developed for the factorization of a multivariate polynomial represented by traightline program into its irreducible factors. The algorithm is in random polynomialtime as a function in the input size, total degree, and binary coefficient length for the usual coefficient fields and ..."
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Cited by 29 (8 self)
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An algorithm is developed for the factorization of a multivariate polynomial represented by traightline program into its irreducible factors. The algorithm is in random polynomialtime as a function in the input size, total degree, and binary coefficient length for the usual coefficient fields and outputs a straightline program, which with controllably high probability correctly determines the irreducible factors. It also returns the probably correct multiplicities of each distinct factor. If th oefficient field has finite characteristic p and p divides the multiplicities of some irreducible factors our algorithm constructs straightline programs for the appropriate pth powers of such factors. Also a probabilistic algorithm is presented that allows to convert a polynomial given by a straightline program into its sparse representation. This conversion algorithm is in randompolynomial time in the previously cited parameters and in an upper bound for the number of nonzero...
Checking polynomial identities over any field: Towards a derandomization
 In Proceedings of the 30th Annual ACM Symposium on Theory of Computing
, 1998
"... We present a Monte Carlo algorithm for testing multivariate polynomial identities over any field using fewer random bits than other methods. To test if a polynomial P (x 1�::: �xn) is zero, our method uses Pn i=1dlog(di +1)erandom bits, where di is the degree of xi in P, to obtain any inverse polyno ..."
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Cited by 22 (0 self)
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We present a Monte Carlo algorithm for testing multivariate polynomial identities over any field using fewer random bits than other methods. To test if a polynomial P (x 1�::: �xn) is zero, our method uses Pn i=1dlog(di +1)erandom bits, where di is the degree of xi in P, to obtain any inverse polynomial error in polynomial time. The algorithm applies to polynomials given as a black box or in some implicit representation such as a straightline program. Our method works by evaluating P at truncated formal power series representing square roots of irreducible polynomials over the field. This approach is similar to that of Chen and Kao [CK97], but with the advantage that the techniques are purely algebraic and apply to any field. We also prove a lower bound showing that the number of random bits used by our algorithm is essentially optimal in the blackbox model. 1
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 18 (10 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.
The Complexity of Quantifier Elimination and Cylindrical Algebraic Decomposition
 Proceedings ISSAC 2007
, 2007
"... This paper has two parts. In the first part we give a simple and constructive proof that quantifier elimination in real algebra is doubly exponential, even when there is only one free variable and all polynomials in the quantified input are linear. The general result is not new, but we hope the simp ..."
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Cited by 16 (8 self)
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This paper has two parts. In the first part we give a simple and constructive proof that quantifier elimination in real algebra is doubly exponential, even when there is only one free variable and all polynomials in the quantified input are linear. The general result is not new, but we hope the simple and explicit nature of the proof makes it interesting. The second part of the paper uses the construction of the first part to prove some results on the effects of projection order on CAD construction — roughly that there are CAD construction problems for which one order produces a constant number of cells and another produces a doubly exponential number of cells, and that there are problems for which all orders produce a doubly exponential number of cells. The second of these results implies that there is a true singly vs. doubly exponential gap between the worstcase running times of several modern quantifier elimination algorithms and CADbased quantifier elimination when the number of quantifier alternations is constant.