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) using temporary files /var/folders/40/dy88l5qd361391m_3v2m51bm0000gn/T/M2-44724-0/0PHCinput and /var/folders/40/dy88l5qd361391m_3v2m51bm0000gn/T/M2-44724-0/0PHCoutput o6 = 8 |
i7 : (mv,sv) = mixedVolume(f,StableMixedVolume=>true) using temporary files /var/folders/40/dy88l5qd361391m_3v2m51bm0000gn/T/M2-44724-0/1PHCinput and /var/folders/40/dy88l5qd361391m_3v2m51bm0000gn/T/M2-44724-0/1PHCoutput o7 = (8, 10) o7 : Sequence |
i8 : (mv,q,qsols) = mixedVolume(f,StartSystem=>true); using temporary files /var/folders/40/dy88l5qd361391m_3v2m51bm0000gn/T/M2-44724-0/2PHCinput and /var/folders/40/dy88l5qd361391m_3v2m51bm0000gn/T/M2-44724-0/2PHCoutput |
i9 : q --let's take a look at the start system: 3 5 2 o9 = {(- .960115 - .279607*ii)x y + (- .981459 - .191672*ii)x y + (- .80103 ------------------------------------------------------------------------ 2 2 + .598624*ii)y , (.231295 - .972884*ii)x + (- .813643 - .581364*ii)x*y ------------------------------------------------------------------------ + .981841 + .189708*ii} o9 : List |
i10 : qsols --and its solutions: o10 = {{.857488-.145496*ii, .652327-1.03637*ii}, {-.6364+.453356*ii, ----------------------------------------------------------------------- -.596898+.177232*ii}, {.352494-.757897*ii, .075575+.735741*ii}, ----------------------------------------------------------------------- {-1.02998+1.18398*ii, .904686+.244045*ii}, {-1.07066+.286798*ii, ----------------------------------------------------------------------- -.219861+1.03393*ii}, {.480298-1.17891*ii, -.700569+.538125*ii}, ----------------------------------------------------------------------- {1.2054-.878332*ii, -.61121-.651457*ii}, {-.158645+1.0365*ii, ----------------------------------------------------------------------- .495951-1.04125*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.