RegularChains[SemiAlgebraicSetTools] - Maple Programming Help

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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

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

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

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

Check the sign of a polynomial at that box.

 > p := x^2+y^2-4;
 ${p}{≔}{{x}}^{{2}}{+}{{y}}^{{2}}{-}{4}$ (3)
 > SignAtBox(p, B, R);
 ${0}$ (4)

Checking a couple other signs.

 > SignAtBox(p-1, B, R);
 ${-1}$ (5)
 > SignAtBox(p+1, B, R);
 ${1}$ (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.