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On solutions and Waring's formulas for systems of $ n$ algebraic equations for $ n$ unknowns

Authors: V. R. Kulikov and V. A. Stepanenko
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 26 (2014), nomer 5.
Journal: St. Petersburg Math. J. 26 (2015), 839-848
MSC (2010): Primary 11D72
Published electronically: July 27, 2015
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Abstract | References | Similar Articles | Additional Information

Abstract: A system of $ n$ algebraic equations for $ n$ unknowns is considered, in which the collection of exponents is fixed, and the coefficients are variable. Since the solutions of such systems are $ 2n$-homogeneous, two coefficients in each equation can be fixed, which makes it possible to pass to the corresponding reduced systems. For the reduced systems, a formula for the solution (and also for any monomial of the solution) is obtained in the form of a hypergeometric type series in the coefficients. Such series are represented as a finite sum of Horn's hypergeometric series: the ratios of the neighboring coefficients of the latter series are rational functions of summation variables. The study is based on the linearization procedure and on the theory of multidimensional residues. As an application of the main formula, a multidimensional analog is presented of the Waring formula for powers of the roots of the system.

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Additional Information

V. R. Kulikov
Affiliation: Siberian Federal University, Svobodnyi pr. 79, Krasnoyarks 660041, Russia

V. A. Stepanenko
Affiliation: Siberian Federal University, Svobodnyi pr. 79, Krasnoyarks 660041, Russia

Keywords: Algebraic equations, hypergeometric functions, multidimensional logarithmic residue, power sums
Received by editor(s): August 7, 2013
Published electronically: July 27, 2015
Additional Notes: This work was done at Siberian Federal University with the support of the RF Government grant no. 14.Y26.31.0006 for studies under the guidance of leading scientists. Also, the first author was supported by RFBR (grant no. 14-01-31265)
Article copyright: © Copyright 2015 American Mathematical Society