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Factoring Multivariate Polynomials via Partial Differential Equations
 Math. Comput
, 2000
"... A new method is presented for factorization of bivariate polynomials over any field of characteristic zero or of relatively large characteristic. It is based on a simple partial differential equation that gives a system of linear equations. Like Berlekamp's and Niederreiter's algorithms fo ..."
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Cited by 60 (9 self)
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A new method is presented for factorization of bivariate polynomials over any field of characteristic zero or of relatively large characteristic. It is based on a simple partial differential equation that gives a system of linear equations. Like Berlekamp's and Niederreiter's algorithms for factoring univariate polynomials, the dimension of the solution space of the linear system is equal to the number of absolutely irreducible factors of the polynomial to be factored and any basis for the solution space gives a complete factorization by computing gcd's and by factoring univariate polynomials over the ground field. The new method finds absolute and rational factorizations simultaneously and is easy to implement for finite fields, local fields, number fields, and the complex number field. The theory of the new method allows an effective Hilbert irreducibility theorem, thus an efficient reduction of polynomials from multivariate to bivariate.
Lifting and recombination techniques for absolute factorization
 J. Complexity
, 2007
"... Abstract. In the vein of recent algorithmic advances in polynomial factorization based on lifting and recombination techniques, we present new faster algorithms for computing the absolute factorization of a bivariate polynomial. The running time of our probabilistic algorithm is less than quadratic ..."
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Cited by 23 (7 self)
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Abstract. In the vein of recent algorithmic advances in polynomial factorization based on lifting and recombination techniques, we present new faster algorithms for computing the absolute factorization of a bivariate polynomial. The running time of our probabilistic algorithm is less than quadratic in the dense size of the polynomial to be factored.
Computing the radius of positive semidefiniteness of a multivariate real polynomial via a dual of Seidenberg’s method
, 2010
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Factorisation Algorithms for Univariate and Bivariate Polynomials over Finite Fields
, 2004
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The Kharitonov theorem and its applications in symbolic mathematical computation
, 1997
"... this paper we deal with the problem of diagnosing if a polynomial has such behaviour. In the past, various results deriving bounds of root displacement of a complex polynomial from the size of perturbations of the coefficients were published (e.g., the ones cited in (Marden 1966)). Conversely, theor ..."
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this paper we deal with the problem of diagnosing if a polynomial has such behaviour. In the past, various results deriving bounds of root displacement of a complex polynomial from the size of perturbations of the coefficients were published (e.g., the ones cited in (Marden 1966)). Conversely, theorems elucidating the relationship between coefficient perturbations and root locations are rare, or lead to impractical algorithms in terms of computational costs. One of the few exceptions is the seminal result by V. L. Kharitonov (1978a). He showed that for interval polynomials with real coefficients, it is necessary and sufficient to test just four special members of the polynomial family in order to decide that all polynomials have their roots in the left half of the Gaussian plane (i.e., that they are Hurwitz). He extended his result to complex coefficients in a followup paper; eight test polynomials are required in this case. Sensitivity analysis is an important methodolgy for dealing with symbolic/numeric problem formulations. The inputs are given with imprecise, i.e., floating point coefficients and the algorithms must decide whether within a given perturbation of the coefficients problem instances exist that satisfy the wanted properties. A classical problem is the perturbation of the coefficients to make inconsistent system of linear equations solvable.
Polynomial GCD and factorization via approximate Gröbner bases
, 2010
"... All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. ..."
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All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.
Graphics, Geometry, and Computing
, 1991
"... Computational geometry is a recent discipline with foundations in many branches of mathematics, and which is supposed to serve many applied areas. In this talk, after defining its subject matter, and reviewing its most important tools and paradigms, I will present a quick survey of some recent resul ..."
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Computational geometry is a recent discipline with foundations in many branches of mathematics, and which is supposed to serve many applied areas. In this talk, after defining its subject matter, and reviewing its most important tools and paradigms, I will present a quick survey of some recent results that lie right at the frontiers of computational geometry. I hope this survey will give some idea of how computational geometry relates to those disciplines, and what we can and cannot expect from it.