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36
The Nearest Polynomial with a given zero, Revisited
, 2005
"... In his 1999 Sigsam Bulletin paper [7], H. J. Stetter gave an explicit formula for finding the nearest polynomial with a given zero. This present paper revisits the issue, correcting a minor omission from Stetter’s formula and explicitly extending the results to different polynomial bases. Experiment ..."
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In his 1999 Sigsam Bulletin paper [7], H. J. Stetter gave an explicit formula for finding the nearest polynomial with a given zero. This present paper revisits the issue, correcting a minor omission from Stetter’s formula and explicitly extending the results to different polynomial bases. Experiments with our implementation demonstrate that the formula may not after all, fully solve the problem, and we discuss some outstanding issues: first, that the nearest polynomial with the given zero may be identically zero (which might be surprising), and, second, that the problem of finding the nearest polynomial of the same degree with a given zero may not, in fact, have a solution. A third variant of the problem, namely to find the nearest monic polynomial (given a monic polynomial initially) with a given zero, a problem that makes sense in some polynomial bases but not others, can also be solved with Stetter’s formula, and this may be more satisfactory in some circumstances. This last can be generalized to the case where some coefficients are intrinsic and not to be changed, whereas others are empiric and may safely be changed. Of course, this minor generalization is implicit in [7]; This paper 1 simply makes it explicit.
Tropical Algebraic Geometry in Maple  a preprocessing algorithm for finding common factors to multivariate polynomials with approximate coefficients
, 2009
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Testing polynomial primality with pseudozeros
 In Proceedings of the Fifth Conference on Real Numbers and Computers
, 2003
"... 52, avenue de Villeneuve ..."
Formally reviewed communication A note on a nearest polynomial with a given root
"... In this paper, we consider the problem of a nearest polynomial with a given root in the complex field (the coefficients of the polynomial and the root are complex numbers). We are interested in the existence and the uniqueness of such polynomials. Then we study the problem in the real case (the coef ..."
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In this paper, we consider the problem of a nearest polynomial with a given root in the complex field (the coefficients of the polynomial and the root are complex numbers). We are interested in the existence and the uniqueness of such polynomials. Then we study the problem in the real case (the coefficients of the polynomial and the root are real numbers), and in the realcomplex case (the coefficients of the polynomial are real numbers and the root is a complex number). We derive new formulas for computing such polynomials.
Numerical Analysis: the study of algorithms
, 1998
"... for problems of continuous mathematics whilst in the introduction to [2], R. Loos defines Computer Algebra: that part of computer science which designs, analyzes, implements, and applies algebraic algorithms. Clearly these definitions are similar, and the major distinction is continuous versus algeb ..."
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for problems of continuous mathematics whilst in the introduction to [2], R. Loos defines Computer Algebra: that part of computer science which designs, analyzes, implements, and applies algebraic algorithms. Clearly these definitions are similar, and the major distinction is continuous versus algebraic. Until one introduces a notion of topology, the concepts of algebra and analysis (or continuous mathematics) are entirely independent and complementary. We can therefore expect to find much rich material in their interaction. In comparing these definitions, one really needs to appeal to examples, such as the following, to make the distinctions clearer. Computing the approximate value of a definite integral by GaussChebyshev quadrature is a numerical computation:
1 On the other hand, if your program transforms
"... Numerical Analysis as “the study of algorithms for problems of continuous mathematics”, whilst in the introduction to [2], R. Loos defines Computer Algebra as “that part of computer science which designs, analyzes, implements, and applies algebraic algorithms.” Clearly the definitions are similar, a ..."
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Numerical Analysis as “the study of algorithms for problems of continuous mathematics”, whilst in the introduction to [2], R. Loos defines Computer Algebra as “that part of computer science which designs, analyzes, implements, and applies algebraic algorithms.” Clearly the definitions are similar, and the major distinction is continuous versus algebraic. One really needs to appeal to examples to make the definitions clearer, such as the following. Computing the approximate value of a definite integral by GaussChebyshev quadrature is a Numerical
Towards Factoring Bivariate Approximate Polynomials*
"... A new algorithm is presented for factoring bivariate approximate polynomials over C[a:, y]. Given a particular polynomial, the method constructs a nearby composite polynomial, if one exists, and its irreducible factors. Subject to a conjecture, the time to produce the factors is polynomial in the ..."
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A new algorithm is presented for factoring bivariate approximate polynomials over C[a:, y]. Given a particular polynomial, the method constructs a nearby composite polynomial, if one exists, and its irreducible factors. Subject to a conjecture, the time to produce the factors is polynomial in the degree of the problem. This method has been implemented in Maple, and has been demonstrated to be efficient and numerically robust. 1.
Abstract MAY, JOHN PAUL. Approximate Factorization of Polynomials in Many Variables and Other Problems in Approximate Algebra via Singular Value Decomposition Methods.
"... Aspects of the problem of finding approximate factors of a polynomial in many variables are considered. The idea is that a polynomial may be the result of a computation where a reducible polynomial was expected but due to introduction of floating point coefficients or measurement errors the polyno ..."
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Aspects of the problem of finding approximate factors of a polynomial in many variables are considered. The idea is that a polynomial may be the result of a computation where a reducible polynomial was expected but due to introduction of floating point coefficients or measurement errors the polynomial is irreducible. Introduction of such errors will nearly always cause polynomials to become irreducible. Thus, it is important to be able to decide whether the computed polynomial is near to a polynomial that factors (and hence should be treated as reducible). If this is the case, one would like to be able to find a closest polynomial that does indeed factor. Although this problem is computable
Article electronically published on May 15, 2002 PSEUDOZEROS OF MULTIVARIATE POLYNOMIALS
"... Abstract. The pseudozero set of a system f of polynomials in n complex variables is the subset of Cn which is the union of the zerosets of all polynomial systems g that are near to f in a suitable sense. This concept is made precise, and general properties of pseudozero sets are established. In par ..."
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Abstract. The pseudozero set of a system f of polynomials in n complex variables is the subset of Cn which is the union of the zerosets of all polynomial systems g that are near to f in a suitable sense. This concept is made precise, and general properties of pseudozero sets are established. In particular it is shown that in many cases of natural interest, the pseudozero set is a semialgebraic set. Also, estimates are given for the size of the projections of pseudozero sets in coordinate directions. Several examples are presented illustrating some of the general theory developed here. Finally, algorithmic ideas are proposed for solving multivariate polynomials. 1.