Abstract
Given a multivariate polynomial F(x, y, ...,z), this paper deals with calculating the roots ofF w.r.t.x in terms of formal power series or fractional-power series iny, ...,z. If the problem is regular, i.e. the expansion point is not a singular point of a root, then the calculation is easy, and the irregular case is considered in this paper. We extend the generalized Hensel construction slightly so that it can be applied to the irregular case. This extension allows us to calculate the roots of bivariate polynomial F(x, y) in terms of Puiseux series iny. For multivariate polynomial F(x, y, ...,z), we consider expanding the roots into fractional-power series w.r.t. the total-degree ofy, ...,z, and the roots are expressed in terms of the roots of much simpler polynomials.
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Work supported in part by Japanese Ministry of Education, Science and Culture under Grants 03558008 and 04245102.
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Sasaki, T., Kako, F. Solving multivariate algebraic equation by Hensel construction. Japan J. Indust. Appl. Math. 16, 257–285 (1999). https://doi.org/10.1007/BF03167329
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DOI: https://doi.org/10.1007/BF03167329