On Wed, May 20, 2009 at 12:20 AM, deutsch <deutsch@duke.poly.edu> wrote:

I am interested in an analytic device to handle the following simple situation. I have a polynomial P in several variables, x,y,z,u, say. It is of degree 2 in each variable. I would like to evaluate the polynomial when x^2 = 0, y^2 = 0, z^2 = 0, u^2 = 0 BUT x=y=z=u=1.

I think we may assume commutativity, associativity, distributivity. Here is a simple example with 2 variables: P(x,y) = 2 + 2xy + x^2 + y^2. Letting x^2 = y^2 = 0 and then x = y = 1, the answer is 4. This is the number I am after in this particular example.

Many thanks for any comment, hint, input. Emeric

You've defined a function, let's call it F, from degree-2 polynomials to the reals, defined as the sum of the coefficients of terms of degree 1 or 0 in each variable. I'm not sure what you mean by "an analytic device". This function isn't really "Evaluation of the polynomial" in any normal sense: in particular, it doesn't satisfy F(P)F(Q) = F(PQ). For example, F(x+y) = 2, but F((x+y)(x+y)) = F(x^2 + 2xy + y^2) = 2 != 2 * 2 = F(x + y) F(x + y) It might be more useful if, instead of saying "I'm interested in an analytic device to handle F", you gave some indication of why you find F interesting, what use you would like to put F to, or what properties of F you are interested in or would like to try to prove. Andy Latto Andy.Latto@pobox.com