number of distinct real solutions of a semi-algebraic system - Maple Help

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RegularChains[SemiAlgebraicSetTools][RealRootCounting] - number of distinct real solutions of a semi-algebraic system

Calling Sequence

RealRootCounting(F, N, P, H, R)

Parameters

R

-

polynomial ring

F

-

list of polynomials of R

N

-

list of polynomials of R

P

-

list of polynomials of R

H

-

list of polynomials of R

Description

• 

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.

Examples

withRegularChains:

withSemiAlgebraicSetTools:

R:=PolynomialRingy,x:

F:=x21,y2+2xy+1

F:=x21,2xy+y2+1

(1)

Compute the number of nonnegative solutions.

N:=x,y;P:=;H:=

N:=x,y

P:=

H:=

(2)

RealRootCountingF,N,P,H,R

0

(3)

R:=PolynomialRingc,z,y,x

R:=polynomial_ring

(4)

F:=1cxxy2xz2,1cyyx2yz2,1czzx2zy2,8c6+378c327

F:=xy2xz2cx+1,x2yyz2cy+1,x2zy2zcz+1,8c6+378c327

(5)

Require c to be positive here.

N:=;P:=c;H:=

N:=

P:=c

H:=

(6)

RealRootCountingF,N,P,H,R

4

(7)

See Also

ComplexRootClassification, RealRootClassification, RealRootIsolate , RegularChains


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