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Date: 13-2-2019
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Date: 17-2-2019
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Date: 23-2-2019
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The Sylvester matrix can be implemented in the Wolfram Language as:
SylvesterMatrix1[poly1_, poly2_, var_] :=
Function[{coeffs1, coeffs2}, With[
{l1 = Length[coeffs1], l2 = Length[coeffs2]},
Join[
NestList[RotateRight, PadRight[coeffs1,
l1 + l2 - 2], l2 - 2],
NestList[RotateRight, PadRight[coeffs2,
l1 + l2 - 2], l1 - 2]
]
]
][
Reverse[CoefficientList[poly1, var]],
Reverse[CoefficientList[poly2, var]]
]
For example, the Sylvester matrix for and
is
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The determinant of the Sylvester matrix of two polynomials is the resultant of the polynomials.
SylvesterMatrix is an (undocumented) method for the Resultant function in the Wolfram Language (although it isdocumented in Trott 2006, p. 29).
REFERENCES:
Akritas, A. G. "Sylvester's Forgotten Form of the Resultant." Fib. Quart. 31, 325-332, 1993.
Akritas, A. G. "Sylvester's Form of the Resultant and the Matrix-Triangularization Subresultant prs Method." Proceedings of the Conference on Computer Aided Proofs in Analysis, Cincinnati, Ohio, March, 1989 (Ed. K. R. Meyer and D. S. Schmidt.) IMA Volumes in Mathematics and its Applications, 28, 5-11, 1991.
Laidacker, M. A. "Another Theorem Relating Sylvester's Matrix and the Greatest Common Divisor." Math. Mag. 42, 126-128, 1969.
Trott, M. The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, p. 28, 2006. http://www.mathematicaguidebooks.org/.
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