Bernstein–Kushnirenko theorem

Remarks

Bernstein's proof deforms the system along a curve and studies the solutions as Puiseux series; the leading exponents of these series are governed by the faces of the Newton polytopes. Bernstein also gave an explicit sufficient condition on the coefficients, stated with initial forms. Suppose that, for every nonzero $w \in \R^n$, the initial forms of $f_1, \dots, f_n$ in direction $w$ have no common zero in $(\C^*)^n$. Then all common zeros in $(\C^*)^n$ are isolated, and their number with multiplicity equals $\operatorname{MV}(P_1, \dots, P_n)$. Take $n = 1$ and $f = a_0 + a_1 x + \cdots + a_d x^d$ with $a_0 a_d \ne 0$. The Newton polytope is $[0, d]$, of length $d$, and the theorem says $f$ has $d$ roots in $\C^*$ counted with multiplicity. For two bilinear equations $a + bx + cy + dxy = 0$ and $a' + b'x + c'y + d'xy = 0$, both polytopes are the unit square, so the generic number of solutions in $(\C^*)^2$ is $2! \cdot 1 = 2$. Let $f_i$ be a generic dense polynomial of degree $d_i$, so that $P_i$ is $d_i$ times the standard simplex. Then the mixed volume is $d_1 d_2 \cdots d_n$, and the bound agrees with Bézout's theorem. For sparse systems the mixed volume can be much smaller than the Bézout number, which is why the theorem is useful. The count concerns only zeros in the torus: zeros with some coordinate equal to $0$ are not counted.

Going deeper

For a student

The theorem is the sparse refinement of Bézout's theorem: it sees which monomials actually occur. It is central in toric geometry, where the mixed volume becomes an intersection number of divisors on a toric variety attached to the polytopes. In tropical geometry, the same count appears as the number of intersection points of tropical hypersurfaces, counted with lattice multiplicities. In numerical algebraic geometry, polyhedral homotopy methods start from the mixed volume to track exactly the generic number of solutions. Computing the mixed volume uses mixed subdivisions of the Minkowski sum $P_1 + \cdots + P_n$.