RegularChains[SemiAlgebraicSetTools]
RealRootCounting
number of distinct real solutions of a semi-algebraic system
Calling Sequence
Parameters
Description
Examples
RealRootCounting(F, N, P, H, R)
R
-
polynomial ring
F
list of polynomials of R
N
P
H
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.
withRegularChains:
withSemiAlgebraicSetTools:
R≔PolynomialRingy,x:
F≔x2−1,y2+2xy+1
F≔x2−1,2xy+y2+1
Compute the number of nonnegative solutions.
N≔x,y;P≔;H≔
N≔x,y
P≔
H≔
RealRootCountingF,N,P,H,R
0
R≔PolynomialRingc,z,y,x
R≔polynomial_ring
F≔1−cx−xy2−xz2,1−cy−yx2−yz2,1−cz−zx2−zy2,8c6+378c3−27
F≔−xy2−xz2−cx+1,−yx2−yz2−cy+1,−zx2−zy2−cz+1,8c6+378c3−27
Require c to be positive here.
N≔;P≔c;H≔
N≔
P≔c
4
See Also
ComplexRootClassification
RealRootClassification
RealRootIsolate
RegularChains
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