Is there any theorem that tell us the number of complex roots of a system of multi variable polynomial equations? in single variable case, we have Rouche's is applicable. but is any extension to theorem for multi-variate polynomial equation ?
It depend on what you call a solution ! If all polynomials are homogeneous, then the number of projective solution counted with multiplicities is the product of the degree of the equations (Bézout bound) for instance.
Thank you oliver, what if the all polynomials are not homogeneous ? Is any theorem that discuss about the solutions of system of polynomial equations?
The problem is the geometric structure where you look for solutions. For affine space, homogeneisation of polynomials (adding a variable to put every monomial of a polynomial of the same degree) prove you that Bézout is an upper bound. You can look for toric solutions (over a torus) and better bounds exists (BKK bounds for instance) ...
If you look in affine plane, take a look to the wonderful book of Cox, Little and O'Shea "Ideals, varieties and algorithms". It is pretty friendly introduction to algebraic geometry and cover this aspect.
thank you again olivier,
but I didnt exactly understand what do you mean by affine plane ? polynomials are nonlinear ! so i wonder what space do u call affine ?
For instance R^2 is the affine real plane, C^2 the affine complex plane. What I said before is true over C^n for n equations defining a finite number of solutions. In this case, the number of solutions (if finite) is less or equal to the product of the degree of the equations. Over the real, i.e. in R^n the bound is less interesting because the number of real solution is more sophisticated to bound in an interesting way.
This is because C^n is a subset of the projective space P^n(C) and the Bézout bound occur exactly in P^n(C).
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