Can Mathematica factor multivariate polynomials with 4 or more variables? And with high degrees (>10)

The Mathematica documentation clearly states

the Wolfram Language routinely factors degree-100 polynomials in 3 variables

I’m interested in factoring systems of polynomials in as many as 10 or 20 variables. The systems I have are sparse in the sense that if there are 20 variables then likely no more than 3-6 variables will appear per equation and there will likely only be two terms per equation. I’ve used Solve to test on some small systems with success. To be clear, we don’t actually need to factor the polynomials necessarily. If we factor them, then we have what we need. What we really want are the roots to the system with respect to the symbolic coefficients. Solve worked for some small test systems. Factor would also work for a single polynomial, but we need a system of polynomials and I don’t see that Factor will take a system.

I’ve looked at papers by searching scholar.google.com and it seems our problem is a solved problem in mathematics. Algorithms seem to exist for such a problem, but I’m even unclear on this since the papers are too densely packed with math for me to easily understand.

Any help would be appreciated.