Author |
Topic: polynomial function fixations (Read 761 times) |
|
william wu
wu::riddles Administrator
Gender: 
Posts: 1291
|
 |
polynomial function fixations
« on: Aug 29th, 2003, 10:34pm » |
 Quote  Modify
|
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?
|
|
 IP Logged |
[ wu ] : http://wuriddles.com / http://forums.wuriddles.com
|
|
|
Icarus
wu::riddles Moderator
Uberpuzzler
Boldly going where even angels fear to tread.
Gender:
Posts: 4863
|
|
Re: polynomial function fixations
« Reply #1 on: Sep 5th, 2003, 4:01pm » |
Quote Modify
|
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 |
"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
|
|
|
|
RIDDLES SITE
WRITE MATH!
Home 
Help 
Search 
Members 
Login 
Register
Reply
Notify of replies
Send Topic
Print