How many roots exist of a system of polynomials

Question

For normal polynomials we have $$\sum_{i=0}^n a_ix^{n-i} =0 $$ and for system of polynomials I will write as such $$\sum_{i_1=0}^{n_1}\sum_{i_2}^{n_2}...\sum_{i_m}^{n_m}a_{1,i_1,i_2...i_m}x_1^{n_1-i_1}x_2^{n_2-i_2}...x_m^{n_m-i_m}=0$$ $$\sum_{i_1=0}^{n_1}\sum_{i_2}^{n_2}...\sum_{i_m}^{n_m}a_{2,i_1,i_2...i_m}x_1^{n_1-i_1}x_2^{n_2-i_2}...x_m^{n_m-i_m}=0$$ $$\vdots$$$$\sum_{i_1=0}^{n_1}\sum_{i_2}^{n_2}...\sum_{i_m}^{n_m}a_{m,i_1,i_2...i_m}x_1^{n_1-i_1}x_2^{n_2-i_2}...x_m^{n_m-i_m}=0$$ and so my question would be the maximum number of solutions that could arise be for values of m and n also letting solutions to count by m-tuples $(x_1,x_2...x_m)$ each of these would be a point and each point would count as one solution

Answer 1

$m$ polynomials in $m$ variables of maximal degree $d$ have at most $d^m$ solution points. This number is exact for homogeneous polynomials in projective space. This is a Bezout theorem.

There may be solutions at infinity which reduces the number of solutions in the Cartesian version of the same system. One can capture some of this effect. If the index tuples of the polynomials are inside some convex set each, and if these sets are smaller than the full set of index tuples of the same maximal total degree, then the BKK bound gives a tighter bound for the number of solution points. The computation of this bound via Minkowski sum of the convex set and determination of its volume is a combinatorial task with quickly raising complexity.