The mixed volume of a polynomial system S:={f1,…,fn} is defined as follows: Let P1,…,Pn be the Newton polytopes of f1,…,fn, i.e., Pi is the convex hull of the exponents of the monomials in the support of fi. The mixed volume of S is
∑1≤h≤n ∑1≤i1…≤ih≤n (-1)n-hVn(Pi1+…+Pih),
where Vn denotes the n-dimensional Euclidean volume.Bernstein’s theorem (D. N. Bernstein,The number of roots of a system of equations, Functional. Anal. Appl 9 (1975), no. 3, 183-185), a generalization of the classical Bezout’s theorem, shows that for a zero-dimensional system, the mixed volume provides an upper bound on the number of complex isolated roots. If the coefficients of the system are sufficiently generic, the mixed volume is a sharp bound.
i1 : R = CC[x,y]; |
i2 : f = { x^3*y^5 + y^2 + x^2*y, x*y + x^2 - 1}; |
i3 : I=ideal f; o3 : Ideal of R |
i4 : dim I -- warning: experimental computation over inexact field begun -- results not reliable (one warning given per session) o4 = 0 |
i5 : degree I o5 = 10 |
i6 : m = mixedVolume(f) -- counts the number of complex roots in the torus (without zero components) o6 = 8 |
i7 : (mv,sv) = mixedVolume(f,StableMixedVolume=>true) o7 = (8, 10) o7 : Sequence |
i8 : (mv,q,qsols) = mixedVolume(f,StartSystem=>true); |
i9 : q --let's take a look at the start system: 3 5 2 o9 = {(.536639 - .843812*ii)x y + (- .960158 + .279458*ii)x y, (.746596 - ------------------------------------------------------------------------ 2 .665278*ii)x + (.992313 + .123751*ii)x*y + .704778 - .709428*ii} o9 : List |
i10 : qsols --and its solutions: o10 = {{-.296956+.665448*ii, 1.02829-.337746*ii}, {.252808+.610316*ii, ----------------------------------------------------------------------- .126444+1.10198*ii}, {.017880-.581105*ii, -.967054-.613442*ii}, ----------------------------------------------------------------------- {-.10855-1.53838*ii, .745355+.499693*ii}, {.499752-1.01646*ii, ----------------------------------------------------------------------- -.42924+.869117*ii}, {.480218+1.30259*ii, -.914167+.114052*ii}, ----------------------------------------------------------------------- {-.313972+1.42018*ii, -.240087-.878372*ii}, {-.53118-.862593*ii, ----------------------------------------------------------------------- .650459-.755284*ii}} o10 : List |
Note that only those solutions with nonzero components are shown, even if StableMixedVolume is true. See the end of the temporary output file for the solutions with zero components.