Hi r.math. I'm interested in taking a given system of polynomials, and turning it into a parameterized system. For example, the ideal of basic facts about a resistor: < v - i*r = 0, i - v*g = 0, p - i*v = 0 >
I would like to turn it into the following system of functions given any two variables. For example:
(r, g, v, i, p) in terms of r and v would be (r, 1/r, v, v/r, v^2 / r ) (r, g, v, i, p) in terms of g and p would be (1/g, g, sqrt(p/g), sqrt(p*g), p)
I can tell that there should be two parameters because there are five variables and three equations, but another system could use a different number.
Also I recognize that there are some sub-situations where one of the variables is 0. For example if v is 0, that leaves only this possible situation: (r, g, 0, 0, 0) since you can see that v, i, and p are all simultaneous 0s and r and g are no longer linked to each other.
I have experience with Groebner bases and linear algebra, but was wondering if anyone had a more solid (less guess-based) approach that I could put into an algorithm. Thanks all.