RealRootCounting - Maple Help

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

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{SemiAlgebraicSetTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[y,x\right]\right):$
 > $F≔\left[{x}^{2}-1,{y}^{2}+2xy+1\right]$
 ${F}{≔}\left[{{x}}^{{2}}{-}{1}{,}{2}{}{x}{}{y}{+}{{y}}^{{2}}{+}{1}\right]$ (1)

Compute the number of nonnegative solutions.

 > $N≔\left[x,y\right];$$P≔\left[\right];$$H≔\left[\right]$
 ${N}{≔}\left[{x}{,}{y}\right]$
 ${P}{≔}\left[\right]$
 ${H}{≔}\left[\right]$ (2)
 > $\mathrm{RealRootCounting}\left(F,N,P,H,R\right)$
 ${0}$ (3)
 > $R≔\mathrm{PolynomialRing}\left(\left[c,z,y,x\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (4)
 > $F≔\left[1-cx-x{y}^{2}-x{z}^{2},1-cy-y{x}^{2}-y{z}^{2},1-cz-z{x}^{2}-z{y}^{2},8{c}^{6}+378{c}^{3}-27\right]$
 ${F}{≔}\left[{-}{x}{}{{y}}^{{2}}{-}{x}{}{{z}}^{{2}}{-}{c}{}{x}{+}{1}{,}{-}{y}{}{{x}}^{{2}}{-}{y}{}{{z}}^{{2}}{-}{c}{}{y}{+}{1}{,}{-}{z}{}{{x}}^{{2}}{-}{z}{}{{y}}^{{2}}{-}{c}{}{z}{+}{1}{,}{8}{}{{c}}^{{6}}{+}{378}{}{{c}}^{{3}}{-}{27}\right]$ (5)

Require c to be positive here.

 > $N≔\left[\right];$$P≔\left[c\right];$$H≔\left[\right]$
 ${N}{≔}\left[\right]$
 ${P}{≔}\left[{c}\right]$
 ${H}{≔}\left[\right]$ (6)
 > $\mathrm{RealRootCounting}\left(F,N,P,H,R\right)$
 ${4}$ (7)