Factoring multivariate polynomials over algebraic number fields
SIAM Journal on Computing
Approximate solutions of polynomial equations
Journal of Symbolic Computation
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
Modern Computer Algebra
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Let F(x,y) be a polynomial over a field K and m a nonnegative integer. We call a polynomial g over K an m-near solution of F(x,y) if there exists a c@?K such that F(x,g)=cx^m, and the number c is called an m-value of F(x,y) corresponding to g. In particular, c can be 0. Hence, by viewing F(x,y)=0 as a polynomial equation over K[x] with variable y, every solution of the equation F(x,y)=0 in K[x] is also an m-near solution. We provide an algorithm that gives all m-near solutions of a given polynomial F(x,y) over K, and this algorithm is polynomial time reducible to solving one variable equations over K. We introduce approximate solutions to analyze the algorithm. We also give some interesting properties of approximate solutions.