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.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 3 5 2 2 2 o3 = ideal (x y + x y + y , x + x*y - 1) 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 /tmp/M2-5174-1PHCinput and /tmp/M2-5174-1PHCoutput o6 = 8 |
i7 : (mv,sv) = mixedVolume(f,stableMV=>true) using temporary files /tmp/M2-5174-2PHCinput and /tmp/M2-5174-2PHCoutput o7 = (8, 10) o7 : Sequence |
i8 : (mv,q,qsols) = mixedVolume(f,startSystem=>true); using temporary files /tmp/M2-5174-3PHCinput and /tmp/M2-5174-3PHCoutput |
i9 : q --let's take a look at the start system: 3 5 2 o9 = {(.728837 - .684688*ii)x y + (.973828 + .227286*ii)x y + (- .976956 + ------------------------------------------------------------------------ 2 2 1.74515*ii)y , (.44653 - .894768*ii)x + (- .617445 - .786614*ii)x*y + ------------------------------------------------------------------------ 1.47976 + 1.34547*ii} o9 : List |
i10 : qsols --and its solutions: o10 = {{.646817-.392041*ii, .597831+.070632*ii}, {-.537105+.902622*ii, ----------------------------------------------------------------------- .429286-.626667*ii}, {.0288432-.256438*ii, .824626+3.80858*ii}, ----------------------------------------------------------------------- {-1.14526+.953079*ii, .771425+.289597*ii}, {-1.15438+.260242*ii, ----------------------------------------------------------------------- -.114404+.885928*ii}, {.82435-1.20233*ii, -.962744+.262992*ii}, ----------------------------------------------------------------------- {1.38479-.511964*ii, -.389985-.906534*ii}, {-.0480499+.246829*ii, ----------------------------------------------------------------------- -1.15603-3.78453*ii}} o10 : List |
Note that only those solutions with nonzero components are shown, even if stableMV is true. See the end of the temporary output file for the solutions with zero components.