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.

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$

$\mathrm{with}\left(\mathrm{SemiAlgebraicSetTools}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[y\,x\right]\right)\:$

$F≔\left[x^2-1\,y^2+2xy+1\right]$

$F≔\left[x^2-1\,2xy+y^2+1\right]$

$N≔\left[x\,y\right]\;$$P≔\left[\right]\;$$H≔\left[\right]$

$N≔\left[x\,y\right]$

$P≔\left[\right]$

$H≔\left[\right]$

$\mathrm{RealRootCounting}\left(F\,N\,P\,H\,R\right)$

$0$

$R≔\mathrm{PolynomialRing}\left(\left[c\,z\,y\,x\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$F≔\left[1-cx-x{y}^2-x{z}^2\,1-cy-y{x}^2-y{z}^2\,1-cz-z{x}^2-z{y}^2\,8c^6+378c^3-27\right]$

$F≔\left[-x{y}^2-x{z}^2-cx+1\,-y{x}^2-y{z}^2-cy+1\,-z{x}^2-z{y}^2-cz+1\,8c^6+378c^3-27\right]$

$N≔\left[\right]\;$$P≔\left[c\right]\;$$H≔\left[\right]$

$N≔\left[\right]$

$P≔\left[c\right]$

$H≔\left[\right]$

$\mathrm{RealRootCounting}\left(F\,N\,P\,H\,R\right)$

$4$