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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 57 (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 22 (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.
Graphics, Geometry, and Computing
"... ABSTRACT 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 rec ..."
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ABSTRACT 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. 1 What is computational geometry? Computational geometry (CG) is usually defined as &quot;the mathematical study of algorithms for solving geometric problems.&quot; Like most definitions, this one seems fine, but only until one really needs it. When I started preparing this talk I soon realized that this definition was too vague, and could easily be stretched to include most of mathematics an computer science. So, to make my task feasible, I must &quot;clarify &quot; that definition, by imposing four additional conditions. First, consider the phrase &quot;geometric problem. &quot; Obviously, any mathematical problem can be reformulated as a geometric problem in rather trivial ways (&quot;How many triangles is two triangles plus two triangles?&quot;). So, Condition One is that geometry must play an essential rolein the statement of the problem, or in the description of the algorithm, or in the accompanying proofs and analyses. Thus, for example, &quot;compute the convex hull of n given points &quot; easily passes this criterion; &quot;sort n given points in lefttoright order &quot; probably does not. Second, we need to clarify the meaning of &quot;algorithm. &quot; You may have noticed that in its origins, and throughout most of its history, geometry was primarily a computational discipline. To the ancient Greeks, ruler and compass were not artists ' tools, but rather computing
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.