Vol. 43, No. 3, 1972

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Invariant functions of an iterative process for maximization of a polynomial

Peter F. Stebe

Vol. 43 (1972), No. 3, 765–783

Let P be a polynomial with real non-negative coefficients and variables xij,i = 1,,k,j = 1,,ni. Let d = k1ni. Let Rd be the d-dimensional real vector space. Let M be the subset of Rd defined by

M˜ = {x|x ∈ Rd,xi,j ≧ 0,n   xi,j = 1}

where the symbols xi,j denote the components of x. If x is a vector in the interior of M, define τ(x) as the vector in M with components xi,jgiven by

         x  ∂∂Px-
x′i,j = ∑i--i,j--ij----.
h=1x  ∂∂xPi,h n

The expression on the right is evaluated at x. The transformation τ is defined on the boundary of M by the same formula if the denominators do not vanish.

Let F be the set of fixed points of τ in M. It is shown that if τ is a homeomorphism of M onto itself, there is a set of dk functions f1, , fdk defined on MF such that fi(x) = fi(τ(x)) for x MF. The functions fi are continuous and independent on an open dense subset of MF. Explicit expressions for certain invariant functions are also obtained.

Mathematical Subject Classification 2000
Primary: 58E15
Secondary: 54H15
Received: 18 February 1971
Revised: 20 July 1972
Published: 1 December 1972
Peter F. Stebe