More on solving systems of power equations

Wu, Yingquan

doi:10.1090/S0025-5718-10-02363-X

More on solving systems of power equations
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by Yingquan Wu;

Math. Comp. 79 (2010), 2317-2332

DOI: https://doi.org/10.1090/S0025-5718-10-02363-X

Published electronically: April 20, 2010

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Abstract:

It is known that a system of power equations can be reduced to a single-variable polynomial equation by exploiting the so-called Newton’s identities. In this work, we investigate four new types of power equation systems. In the first two types we allow the powers to be a mix of positive and negative terms, whereas in the literature the system of power equations involves only positive powers. The first type involves only positive signs of powers, whereas the second type expands to involve both positive and negative signs. We present algebraic methods to solve the system and furthermore fully characterize the number of nontrivial solutions. The other two types are defined over finite fields and otherwise are the same as the conventional system of power equations. The methodology for solving the third type can be viewed as a generalization of the Berlekamp algorithm. The solution space of the last system is fully characterized despite the fact that the number of equations is two less than the number of unknowns.

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