Enumerative Real Algebraic Geometry: An Example

Given a polytope P with vertices in the integral lattice, what are the possible numbers of real solutions to systems with that Newton polytope? We shall focus on understanding r(P), the maximum number of real solutions to a sparse system with Newton polytope P. The following example serves as an introduction to this question of the possible numbers of real solutions to a polynomial system.

Example 2.3 Let d be a positive integer, and set

P_d := Convex hull {(0,0,0), (1,0,0), (0,1,0), (1,1,d)} , and

Q_d := Convex hull {(0,0,0), (1,0,0), (0,1,0), (0,0,d)} .

(Figure 1 shows P₄ and Q₄.) These tetrahedrons each have normalized volume d.

A general sparse system with Newton polytope P_d

A₁xyz^d + B₁x + C₁y + D₁	=	0
A₂xyz^d + B₂x + C₂y + D₂	=	0
A₃xyz^d + B₃x + C₃y + D₃	=	0

is equivalent to a system of the form

z^d = a, x = b, y = c,

where a, b, c are in R. Thus the original system has d complex solutions, but only 0, 1, or 2 real solutions, so r(P_d) = d.

A general sparse system with Newton polytope Q_d consists of 3 polynomials of the form

Ax + By + C(z) ,

where C(z) has degree d. This is equivalent to a system of the form

f(z) = 0, x = g(z), y = h(z)

here f(z), g(z), and h(z) are real polynomials with d = deg f(z), but d-1 = deg g(z) = deg h(z). In this case, the general system again has d complex solutions, but there are systems with any number of real solutions, so r(Q_d) = d.

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2.i.b. An Example