How to determine that a set of equations has a solution or infinite many solution or no solution?

Question

I have a set of polynomials (the variables are $a,b,c,d,e,f,g,h,i,j,a_1,b_1,c_1,d_1,e_1,f_1,g_1,h_1,i_1,j_1$, the polynomials are $L_i$, $i=1,\ldots,20$): $$ aa_1=0 ; [L_1]\\ e a_1 + b e_1=0; [L_2]\\ b b_1=0; [L_{3}]\\ h a_1 + c h_1 + f e_1=0; [L_{4}]\\ f b_1 + c f_1=0; [L_{5}]\\ c c_1=0; [L_{6}]\\ j a_1 + d j_1 + i e_1 + g h_1=0; [L_{7}]\\ i b_1 + d i_1 + g f_1=0; [L_{8}]\\ g c_1 + d g_1=0; [L_{9}]\\ d d_1=0; [L_{10}]\\ j_1=0; [L_{11}]\\ i_1=0; [L_{12}]\\ g_1=0; [L_{13}]\\ d_1=0; [L_{14}]\\ b c j - b g h - c e i + e f g=0; [L_{15}]\\ h i-f j=0; [L_{16}]\\ j=0; [L_{17}]\\ h_1 i_1-f_1 j_1=0; [L_{18}]\\ b_1 c_1 j_1 - b_1 g_1 h_1 - c_1 e_1 i_1 + e_1 f_1 g_1=0; [l_{19}]\\ j a_1 c_1 i_1 - i b_1 c_1 j_1 - j a_1 f_1 g_1 + i b_1 g_1 h_1 + i c_1 e_1 i_1 - i e_1 f_1 g_1=0; [L_{20}] $$

I would like to know that the set of equations: $L_i$, $i=1,\ldots,20$, has a solution or infinite many solution or no solution. Are there some software can do this job? Thank you very much.

Answer 1

If all the letters denoted indeterminates, then you can easily feed the system into Macaulay2, for example.

It will then turn out that the ideal generated by the polynomials $f_i$ is $9$-dimensional, so there are certainly many (possibly complex) solutions. In particular, $0$ is in the solution set, so you always have at least one real solution.

Answer 2

This is probably doable without any software. Start by substituting the variables that you know are $0$. Then you've also got a bunch of equations of the form $xy = 0$; make case distinctions $x = 0$, $y = 0$ and see what you wind up with. It wouldn't surprise me if what remains (in each of the many cases) is easy enough to solve.