# Geometry g.algebraica – Bernstein linked to the number of roots of a multivariable rectangular polynomial system

I would like to know what is the Bernstein limit in multivariate polynomial systems where there are more equations than unknowns. I have a multivariable polynomial system of zero dimension "generic" with a set of scattered real coefficients:
$$begin {pmatrix} p_1 (x_1, ldots, x_d) \ vdots \ p_m (x_1, ldots, x_d) end {pmatrix} = 0,$$
where $$m> d$$. When $$m = d$$, you can definitely limit the number of zeros by the mixed volume linked to BKK, but I'm not sure the limit continues in the $$m> d$$ (And if he does what seems the limit).

My motivating polynomial system comes from the analysis of a truss / graphic (with adjacency matrix $$A = (a_ {ij})$$), and after some trigonometric identities I end with the polynomial system:
begin {aligned} sum_ {j = 2} ^ d a_ {1j} x_j & = 0 \ a_ {i1} x_i + sum_ {j = 2} ^ d a_ {ij} x_iy_j – sum_ {j = 2} ^ d a_ {ij} x_jy_i & = 0, 2 leq i leq d, x_i ^ 2 + y_i ^ 2 – 1 & = 0, 2 ≤ i ≤ d. end {aligned}
Here I have $$2d-1$$ equations and $$2d-2$$ variables In a given lattice problem, most of the $$a_ {ij}$$they are zero Assuming that the graph is connected, then I imagine that it could be valid to discard one of the intermediate equations above. However, ideally, you could only apply a limit directly. In practice, I'm interested in the number of real solutions, but I'm happy to have the number of solutions in $$( mathbb {C} ^ *) ^ {2d-2}$$ or $$mathbb {C} ^ {2d-2}$$ to get started.