SignAtBox - Maple Help

RegularChains[SemiAlgebraicSetTools]

 SignAtBox
 return the sign of a polynomial at real point

 Calling Sequence SignAtBox(p, B, R)

Parameters

 p - a polynomial B - a box object encoding a point with real coordinates R - polynomial ring

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{SemiAlgebraicSetTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[y,x\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)

Isolate the real points of a polynomial system and pick one of them.

 > $B≔\mathrm{RealRootIsolate}\left(\left[{x}^{2}-2,y-x\right],\left[\right],\left[x\right],\left[\right],R\right)\left[1\right]$
 ${B}{≔}{\mathrm{box}}$ (2)

Check the sign of a polynomial at that box.

 > $p≔{x}^{2}+{y}^{2}-4$
 ${p}{≔}{{x}}^{{2}}{+}{{y}}^{{2}}{-}{4}$ (3)
 > $\mathrm{SignAtBox}\left(p,B,R\right)$
 ${0}$ (4)

Checking a couple other signs.

 > $\mathrm{SignAtBox}\left(p-1,B,R\right)$
 ${-1}$ (5)
 > $\mathrm{SignAtBox}\left(p+1,B,R\right)$
 ${1}$ (6)

References

 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.

Compatibility

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