Determine the number of solutions of a polynomial system

Question

I have a system of polynomial equations, whose polynomials are all multi linear with coefficients equal to 1.

I want to understand if the system has or not real solutions, and if yes, if they are a finite number or infinite.

I understand that I should calculate the Groëbner basis:

• if it is equal to 1, then I have no solutions.
• What if it is not 1? How can I understand how many solutions I have?
• Is there any other known method to determine the number of solutions?

thanks!

Answer 1

As nobody answers, I will do it by myself :)

In general it is useful to calculate the Groëbner basis of the system given a certain monomial ordering. Once we have the Groëbner basis G, the following conditions are equivalent:

1. For each unknown of the polynomial $x_i$, there is some $m_i ≥ 0$ such that $x_i^{m_i}$ is the leading monomial for some gi ∈ G.
2. The set of the solutions V is finite.

Furthermore, we can also bound the number of solutions by the following result:

$$ |V|\leq \prod m_i $$

where the $m_i$ are the degrees of the leading monomials appearing in G for each unknown $x_i$.

A less strict bound is represented by the Bezout bound, that says that the number of solutions, if finite, is minor or equal to the product of the degrees of the equations of the original system.

This is what I have found; no idea if we can improve these results in some way or if we can find a tight bound without computing the Groëbner basis.

Any further advice is welcome!

PS All the infos have been taken by the book

"Ideals, Varieties, and Algorithms An Introduction to Computational Algebraic Geometry and Commutative Algebra"

Authors: Cox, David A, Little, John, O'Shea, Dona