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August 30th, 2010, 06:45 PM   #1
 
Joined: Aug 2010
Posts: 1
Solutions to Multivariate polynomial equations

Hello everyone,

I got a problem.

I have a bivariate polynomial equations. i.e. f(x,y) with the order n and g(x,y) with the order m. By Bezout's Theorem, there are m*n common solutions. I can use iteration method to get one solution to these equations. The problem is I try to use this solution to change f(x,y) and g(x,y) to f1(x,y) and g1(x,y) respectively. the polynomial f1(x,y) and g1(x,y) have the property that they have all the same solutions that f(x,y) and g(x,y) have but without the calculated solution. i.e. f1(x,y) and g1(x,y) have m*n-1 solutions. then I can use iteration method again to work out the second solution and repeat the procedure until I work out all solutions. these are all in complex domain.

If the above problem is impossible to be solved. then I got a relative one but simple. i.e. I have a bivariate polynomial equations. i.e. f(x,y) with the order n and g(x,y) with the order m, The number of their common real solutions is r. I can use iteration method to get one real solution to these equations. The problem is I try to use this real solution to change f(x,y) and g(x,y) to f1(x,y) and g1(x,y) respectively. the polynomial f1(x,y) and g1(x,y) have the property that they have all the same real solutions that f(x,y) and g(x,y) have but without the calculated real solution. i.e. f1(x,y) and g1(x,y) have r-1 real solutions. then I can use iteration method again to work out the second real solution and repeat the procedure until I work out all real solutions.

Is there any one do some research on relative area or have read some papers about this. Could you please tell me? Thank you very much.

P.S. I know resultant methods, Homotopy method, Newton iteration method, but I don't want to use them, because, they are expensive for high order polynomial equations, for example, order 100 or 1000.

Mingfei
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November 13th, 2011, 09:15 PM   #2
 
Joined: Nov 2011
Posts: 1
Re: Solutions to Multivariate polynomial equations

Hi everyone! I'm working on program mechanics in excel and have two polynomial equations that I'm very curious about the relationship between.

Equation 1 passes through 0,0 ; 1,1 ; 24,64
Equation 2 passes through 0,0 ; 1,1 ; 24,512

As you can tell I've designed the equations so that they always pass through 0,0 and 1,1, but when x=24 I want y=8^2=64 in the first equation and y=8^3=512 in the second equation.

I've deduced what these equations look like (form y = ax^2 + bx + c):

Equation 1: y = (5/69)x^2 + (64/69)x + 0
Equation 2: y = (61/69)x^2 + (8/69)x + 0

I was pleased to find that "a" in both cases had a lowest common denominator of 69, but I can't figure out how the two equations are related. For example, I'd like to understand how you could modify y for x=3 in Equation 2 to make it equal y for x=3 in Equation 1. I've tried plotting out several data points to try to understand the relationship, but to no avail.

Any ideas? Thank you very much for your time!
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