Question: Does there exist an algorithm that solve system of equations with degree at least $2?$
Motivation: Recently I read Shamil Asgarli's article entitled 'A New Proof of Warning's Second Theorem'. The author recap one of Emil Artin's conjectures which was solved by Chevalley.
Theorem: Let $q = p^k$ be a prime power. Suppose that $f_1,...,f_r\in \mathbb{F}_q[x_1,...,x_n]$ are polynomials such that $$n> \sum_{i=1}^r \text{deg } (f_i).$$ Let $Z= \{\textbf{a}\in \mathbb{F}_q^n: f_i(\textbf{a})=0 \text{ for each }i\}$ to be the common zero locus. If $Z\neq \emptyset,$ then $\#Z\geq 2.$
The theorem above is a generalisation of one of the well-known linear algebra facts, that is, given a system of linear equations, if there are more variables than equations and the system has a solution, then it has another solution.
There is an algorithm to obtain solution by using matrix. However, I do not come across any algorithm that give solution of system of equations with degree at least $2.$
I recommend Cramer’s rule. You set up two matrices: One is nXn containing the coefficients of the variables for n simultaneous equations. The other is a 1Xn matrix of the constants that the equations equal. You take a base determinant of nXn ($D_0$) which will be the divisor for the each of the next steps. If $D_0=0$, there is no solution. Then you replace column 1 of the nXn matrix with the values of the 1Xn and take that determinant ($D_1$.) The value $D_1/D_0$ is the new $x^n$ coefficient. You repeat this process for column 2 ($D_2$) to get the value for the $x(^n-1)$ coefficient and continue thru $D_n$ to get the coefficient for $x^1$. Excel (and probably Open Office) has a determinant function MDETERM(). Here is a link and you will see the process for a 3X3 about halfway down.
https://en.wikipedia.org/wiki/Cramer%27s_rule