Conditions for solutions on simultaneous equations

I have a quick question for a problem I am solving:

I am solving a non-linear system of $$4$$ equations with $$4$$ variables, and after brute expansion I have deduced that there are $$24$$ solutions (but only one set of unique values which can be changed to make different solutions). But I am wondering if there is a faster way for non-linear simultaneous equations to determine the number of solutions there are - is there a set for only a certain number of solutions, no solutions, infinite solutions, etc.

Any help is appreciated.

• can anyone help please Commented Jul 19, 2023 at 11:10
• "Non-linear equations" is so broad a categorization that the only possible answer that can be given to you is "Of course not". The only reason we can do this for linear equations is because they are so restrictive. There are other categories of equations where you can make such determinations, but there is not going to be some overall technique that is going to work for every possible set of equations, Commented Jul 20, 2023 at 11:42