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   Author  Topic: polynomial function fixations  (Read 746 times)
william wu
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polynomial function fixations  
« on: Aug 29th, 2003, 10:34pm »
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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?
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Icarus
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Re: polynomial function fixations  
« Reply #1 on: Sep 5th, 2003, 4:01pm »
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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 ai(y)xi  (i <= N)
 
now fixing N values of x, we get N equations
 
[sum]i ai(y)xji = Pj(y)
 
for N polynomials Pj in y. By choosing the xj correctly, we have an independent system. This can be solved giving each ai as a linear combination of the Pj. 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.
« Last Edit: Sep 6th, 2003, 8:46am by Icarus » IP Logged

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