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.