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        riddles >> putnam exam (pure math) >> polynomial function fixations
        
(Message started by: william wu on Aug 29^th, 2003, 10:34pm)

Title: polynomial function fixations
Post by william wu on Aug 29^th, 2003, 10:34pm

Consider a function f : [bbr][times][bbr][to][bbr] such that when you fix x, f(x,y) is a polynomial in y, and when you fix y, f(x,y) is a polynomial in x.

Is f(x,y) a polynomial in both x and y?

Title: Re: polynomial function fixations
Post by Icarus on Sep 5^th, 2003, 4:01pm

[hide]If there exists N such that if for all y, f(x,y) is a polynomial of degree <= N, then

f(x,y) = [sum]_i a_i(y)xⁱ (i <= N)

now fixing N values of x, we get N equations

[sum]_i a_i(y)x_jⁱ = P_j(y)

for N polynomials P_j in y. By choosing the x_j correctly, we have an independent system. This can be solved giving each a_i as a linear combination of the P_j. Thus they must be polynomials themselves.

Therefore if there is an upper limit on the degree of the polynomials of either x or y, then f must be a polynomial of both x and y.

Still need to show that either the limited degree bit is automatic, or else examine what happens when it fails.[/hide]