number of distinct real solutions of a semi-algebraic system
RealRootCounting(F, N, P, H, R)
list of polynomials of R
The command RealRootCounting(F, N, P, H, R) returns the number of distinct real solutions of the system whose equations, inequations, positive polynomials, and non-negative polynomials are given by F, H, P and N respectively.
This computation assumes that the polynomial system given by F and H (as equations and inequations respectively) has finitely many complex solutions.
The base field of R is meant to be the field of rational numbers.
The algorithm is described in the paper by Xia, B., Hou, X.: "A complete algorithm for counting real solutions of polynomial systems of equations and inequalities." Computers and Mathematics with applications, Vol. 44 (2002): pp.633-642.
R ≔ PolynomialRing⁡y,x:
F ≔ x2−1,y2+2⁢x⁢y+1
F ≔ x2−1,2⁢x⁢y+y2+1
Compute the number of nonnegative solutions.
N ≔ x,y;P ≔ ;H ≔
N ≔ x,y
R ≔ PolynomialRing⁡c,z,y,x
R ≔ polynomial_ring
F ≔ 1−c⁢x−x⁢y2−x⁢z2,1−c⁢y−y⁢x2−y⁢z2,1−c⁢z−z⁢x2−z⁢y2,8⁢c6+378⁢c3−27
F ≔ −x⁢y2−x⁢z2−c⁢x+1,−x2⁢y−y⁢z2−c⁢y+1,−x2⁢z−y2⁢z−c⁢z+1,8⁢c6+378⁢c3−27
Require c to be positive here.
N ≔ ;P ≔ c;H ≔
P ≔ c
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