Hostname: page-component-5c6d5d7d68-wpx84 Total loading time: 0 Render date: 2024-08-18T15:52:48.392Z Has data issue: false hasContentIssue false
  • English
  • Français

On relations between Jacobians and resultants of polynomials in two variables

Published online by Cambridge University Press:  17 April 2009

Takis Sakkalis
Takis Sakkalis
Affiliation:
Department of Mathematical, Sciences Oakland University Rochester MI, 48309, United States of America
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper investigates some of the connections between the zeros of a polynomial vector field F = (f, g): ℂ2 ← ℂ2 and the Jacobian determinant J(f, g) of f and g. As a consequence, sufficient conditions are given for F to have no zeros. In addition, in the case where F has an inverse F−1, it is proven that F−1 is also polynomial.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 47 , Issue 3 , June 1993 , pp. 473 - 481
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Adjamagbo, K. and van den Essen, A., ‘A resultant criterion and formula for the inversion of a polynomial map in two variables’, J. Pure Appl. Algebra 64 (1990), 1–6.CrossRefGoogle Scholar
[2]Collins, G.E., ‘The calculation of multivariate polynomial resultants’, J. Assoc. Comput. Mach. 18 (1971), 515–532.CrossRefGoogle Scholar
[3]McKay, J.H. and Wang, S.S-S., ‘An inversion formula for two polynomials in two variables’, J. Pure Appl. Algebra 40 (1986), 245–257.CrossRefGoogle Scholar
[4]McKay, J. H. and Wang, S.S-S., ‘A chain rule for the resultant of two polynomials’, Arch. Math. 53 (1989), 347–351.CrossRefGoogle Scholar
[5]Sard, A., ‘The measure of the critical values of differentiable maps’, Bull. Amer. Math. Soc. 48 (1942), 883–897.CrossRefGoogle Scholar
[6]van der Waerden, B.L., Algebra Vol. 1 (New York, 1970).Google Scholar