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Computing Higher Polynomial Discriminants

Author:

Erich L. KaltofenAuthors Info & Claims

ISSAC '21: Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation

Pages 233 - 239

https://doi.org/10.1145/3452143.3465543

Published: 18 July 2021 Publication History

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Abstract

In https://arxiv.org/abs/1609.00840 (see also https://doi.org/10.1007/s11425-018-1594-2), Dongming Wang and Jing Yang in 2016 have posed the problem how to compute the "third'' discriminant of a polynomial f(x) = (x-α₁)…(x-α_n), δ₃(f) = Π ((α_i+α_j-α_k-αℓ) (α_i-α_j+α_k-αℓ)(α_i-α_j-α_k+αℓ))1≤<j<k<ℓ≤n from the coefficients of f; note that δ₃ is a symmetric polynomial in the α_i. For complex roots, δ₃(f) = 0 if the mid-point (average) of $2$ roots is equal the mid-point of another 2 roots. Iterated resultant computations yield the square of the third discriminant. We apply a symbolic homotopy by Kaltofen and Trager [JSC, vol. 9, nr. 3, pp. 301--320 (1990)] to compute its squareroot. Our algorithm uses polynomially many coefficient field operations in the degree of f.

References

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David Casperson and John McKay. 1994. Symmetric Functions, m-Sets, and Galois Groups. Math. Comp., Vol. 63, 208 (1994), 749--757. http://www.jstor.org/stable/2153295.

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Olga Holtz and Mikhail Tyaglov. 2011. Structured matrices, continued fractions, and root localization of polynomials. CoRR, Vol. 0912.4703 (March 2011). https://arxiv.org/abs/0912.4703.

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E. Kaltofen and B. Trager. 1990. Computing with polynomials given by black boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators. J. Symbolic Comput., Vol. 9, 3 (1990), 301--320. http://users.cs.duke.edu/~elk27/bibliography/90/KaTr90.pdf EKbib/90/KaTr90.pdf.

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Luciano Orlando. 1911. Sul problema di Hurwitz relativo alle parti reali delle radici di un' equazione algebrica. Math. Ann., Vol. 71 (1911), 233--245.

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Dongming Wang and Jing Yang. 2016. The Second Discriminant of a Univariate Polynomial. CoRR, Vol. abs/1609.00840 (Sept. 2016). https://arxiv.org/abs/1609.00840.

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Dongming Wang and Jing Yang. 2019. The Second Discriminant of a Univariate Polynomial. Science China Mathematics (2019). Published Dec. 23, 2019; https://doi.org/10.1007/s11425-018-1594-2.

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Index Terms

Computing Higher Polynomial Discriminants
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Algebraic algorithms
2. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Computations in finite fields
      2. Computations on polynomials

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ISSAC '21: Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation

July 2021

379 pages

ISBN:9781450383820

DOI:10.1145/3452143

General Chair:
Frédéric Chyzak
Inria, France
,
Program Chair:
George Labahn
University of Waterloo, Canada

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Association for Computing Machinery

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Published: 18 July 2021

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ISSAC '21: International Symposium on Symbolic and Algebraic Computation

July 18 - 23, 2021

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Recommendations

Computing polynomial resultants: Bezout's determinant vs. Collins' reduced P.R.S. algorithm

Branching rules for symmetric functions and sln basic hypergeometric series

Symmetric and nonsymmetric Koornwinder polynomials in the q → 0 limit

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