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COMMON FACTORS OF RESULTANTS MODULO p

Part of: General commutative ring theory

Published online by Cambridge University Press:  13 March 2009

DOMINGO GOMEZ ,
JAIME GUTIERREZ ,
ÁLVAR IBEAS  and
DAVID SEVILLA
DOMINGO GOMEZ
Affiliation:
University of Cantabria, E-39071 Santander, Spain (email: domingo.gomez@unican.es)
JAIME GUTIERREZ*
Affiliation:
University of Cantabria, E-39071 Santander, Spain (email: jaime.gutierrez@unican.es)
ÁLVAR IBEAS
Affiliation:
University of Cantabria, E-39071 Santander, Spain (email: alvar.ibeas@unican.es)
DAVID SEVILLA
Affiliation:
Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstraße 69, A-4040 Linz, Austria (email: david.sevilla@oeaw.ac.at)
*
✉For correspondence; e-mail: jaime.gutierrez@unican.es
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Abstract

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We show that the multiplicity of a prime p as a factor of the resultant of two polynomials with integer coefficients is at least the degree of their greatest common divisor modulo p. This answers an open question by Konyagin and Shparlinski.

Keywords

resultantreduction modulo p

MSC classification

Secondary: 13A35: Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 79 , Issue 2 , April 2009 , pp. 299 - 302
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

This work is partially supported by the Spanish Ministry of Education and Science grant MTM2007-67088.

References

[1] Konyagin, S. V. and Shparlinski, I., Character Sums with Exponential Functions and their Applications (Cambridge University Press, Cambridge, 1999).CrossRefGoogle Scholar
[2] Lidl, R. and Niederreiter, H., Finite Fields (Cambridge University Press, Cambridge, 1997).Google Scholar
[3] van der Waerden, B. L., Modern Algebra (Frederick Ungar Publishing Co., New York, 1964).Google Scholar