The command SignAtBox(p, B, R) returns the sign of the polynomial p at the point encoded by the box object B.

The box object B is assumed to be returned by the command RealRootIsolate.

The sign at B of the polynomial p is given as -1, 0, or 1 for negative, null, or positive, respectively.

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

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

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

$R≔\mathrm{polynomial\_ring}$

$B≔{\mathrm{RealRootIsolate}\left(\left[x^2-2\,y-x\right]\,\left[\right]\,\left[x\right]\,\left[\right]\,R\right)}_1$

$B≔\mathrm{box}$

$p≔x^2\+y^2-4$

$p≔x^2+y^2-4$

$\mathrm{SignAtBox}\left(p\,B\,R\right)$

$0$

$\mathrm{SignAtBox}\left(p-1\,B\,R\right)$

$\mathrm{-1}$

$\mathrm{SignAtBox}\left(p\+1\,B\,R\right)$

$1$

F. Boulier, C. Chen, F. Lemaire, M. Moreno Maza "Real root isolation of regular chains." ASCM'2009, Math-for-Industry, Lecture Note Series Vol. 22.

R. Rioboo "Computation of the real closure of an ordered field." ISSAC'92, Academic Press, San Francisco.

The RegularChains[SemiAlgebraicSetTools][SignAtBox] command was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.