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Asymptotes and Intercepts of Real-Power Polynomial Surfaces from the Geometry of the Exponent Polytope

Bruce L. Clarke
SIAM Journal on Applied Mathematics
Vol. 35, No. 4 (Dec., 1978), pp. 755-786
Stable URL: http://www.jstor.org/stable/2100990
Page Count: 32
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Asymptotes and Intercepts of Real-Power Polynomial Surfaces from the Geometry of the Exponent Polytope
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Abstract

A hierarchy of approximations is developed for the real zeros of an arbitrary complex real-power polynomial f(X), X ε Rn. The approximations are obtained by studying the function g(x) defined by g(x) = f(X) (where xi = log |Xi|), called the exponomial of f(X), along rays $x(\lambda) = x_0 + \lambda \hat x$, as λ → ∞. All unbounded real solutions of g(x) = 0 uniformly approach hypersurfaces of asymptotic rays. From the geometry of a convex polytope P, called the exponent polytope of f(X), (a generalization of the Newton polygon) one can determine completely: (1) the limiting form of g(x) along every ray, (2) all the asymptotic rays of the hypersurface g(x) = 0, (3) all branches of g(x) = 0 with a given asymptotic ray, and (4) a hierarchy of asymptotic hypersurfaces giving the global structure of g(x) = 0 for large |x|. These properties of g(x) determine for f(X) = 0: (1) all asymptotes, (2) all intercepts, (3) the complete branching at an intercept and (4) the hierarchy of approximations to the zeros of f(X) within each orthant. The method has been applied to a 2575-term polynomial in 8 variables. The theorems are sometimes applicable when f(X) is an infinite series.

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