I am new to the topic and I am working with the multivariate systems of equations over finite fields. My goal is to solve them. I know that in general the problem of solving such systems is NP-hard. However, there are certain attacks that can solve such systems.
Suppose I work with the first polynomial in my system of $k$ equations in $k$ variables. In my understanding, I can always fix one variable, say $x_0$ and treat the rest as coefficients. That would create a univariate polynomial in $x_0$. Given that I can solve such univariate polynomial, the roots will be functions of the remaining variables. That is $x_0 = f(x_1, \dots, x_k)$. I can then go ahead and substitute $x_0$ in the next polynomial in the system for the expression $f(x_1, \dots, x_k)$. That would reduce the number of variables I am working with by one. If I keep in this fashion, at the end I will get a univariate equation. Suppose that I can solve it, then I solved my system of multivariate equations.
I was wondering if my analysis is incorrect and if I perhaps am describing a well-known attack, then I will highly appreciate any references on this attack.
Thank you!
There is no computationally useful way in which the roots of the univariate polynomial "are functions of the remaining variables". I assume that by "polynomials" you include those of any degree.
Anyway, in order to understand how the solution set is computed, one might want to learn some computational algebraic geometry.