 RegularChains[ParametricSystemTools] - Maple Programming Help

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RegularChains[ParametricSystemTools]

 BorderPolynomial
 compute the border polynomial of a semi-algebraic system

 Calling Sequence BorderPolynomial(F, N, P, H, d, 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 d - positive integer

Description

 • The input is a parametric semi-algebraic system whose parameters are the last d variables of R and whose polynomial equations, non-negative polynomial inequalities, (strictly) positive polynomial inequalities, and polynomial inequations are given respectively by F, N, P, and H.
 • The command BorderPolynomial returns an object of type border_polynomial. It is a list of polynomials of R the product of which is the border polynomial of the input system.
 • If the output border polynomial only contains the parameters, above each parameter value not canceling the border polynomial, the input parametric system has finitely many solutions. Determining conditions on the parameters for the input system to have a prescribed (finite) number of solutions is achieved by the command RealRootClassification.
 • If the input system is not sufficiently generic (and in particular if it is not generically zero-dimensional with respect to the d parameters) then the output is set to a special value, as shown in the examples below.
 • The base field of R is meant to be the field of real numbers. Thus R must be of characteristic zero and must have no parameters (in the sense of the RegularChains library).

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{ParametricSystemTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,s\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $F≔\left[s-\left(y+1\right)x,s-\left(x+1\right)y\right]$
 ${F}{≔}\left[{s}{-}\left({y}{+}{1}\right){}{x}{,}{s}{-}\left({x}{+}{1}\right){}{y}\right]$ (2)
 > $\mathrm{bp}≔\mathrm{BorderPolynomial}\left(F,\left[\right],\left[\right],\left[\right],1,R\right)$
 ${\mathrm{bp}}{≔}{\mathrm{border_polynomial}}$ (3)
 > $\mathrm{Info}\left(\mathrm{bp},R\right)$
 $\left[{s}{,}{s}{+}\frac{{1}}{{4}}\right]$ (4)

The reason why border polynomials must form of type (and cannot just be seen as lists of polynomials) is that under special circumstances, the border polynomial of a parametric semi-algebraic system takes an exceptional value.

The first such case is when the parameters do not appear in the system of polynomials; then there are no border polynomials as in the example below.

 > $R≔\mathrm{PolynomialRing}\left(\left[x,a,b,c\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (5)
 > $\mathrm{bp_nobp}≔\mathrm{BorderPolynomial}\left(\left[{x}^{2}-1\right],\left[\right],\left[\right],\left[\right],3,R\right)$
 ${\mathrm{bp_nobp}}{≔}{\mathrm{border_polynomial}}$ (6)
 > $\mathrm{Info}\left(\mathrm{bp_nobp},R\right)$
 $\left[{1}\right]$ (7)

Another special circumstance is that of overdetermined or inconsistent systems, as in the example below

 > $F≔\left[a{x}^{2}+bx+c,a,b\right]$
 ${F}{≔}\left[{a}{}{{x}}^{{2}}{+}{b}{}{x}{+}{c}{,}{a}{,}{b}\right]$ (8)
 > $\mathrm{bp}≔\mathrm{BorderPolynomial}\left(F,\left[\right],\left[\right],\left[\right],1,R\right)$
 ${\mathrm{bp}}{≔}{\mathrm{border_polynomial}}$ (9)
 > $\mathrm{Info}\left(\mathrm{bp},R\right)$
 $\left[\right]$ (10)

A last special circumstance is when the input system has "generically" infinitely many complex solutions, as in the example below (this is because of the d=2).

 > $F≔\left[a{x}^{2}+bx+c\right]:$$d≔2$
 ${d}{≔}{2}$ (11)
 > $\mathrm{bp}≔\mathrm{BorderPolynomial}\left(F,\left[\right],\left[\right],\left[\right],d,R\right)$
 ${\mathrm{bp}}{≔}{\mathrm{border_polynomial}}$ (12)
 > $\mathrm{Info}\left(\mathrm{bp},R\right)$
 $\left[{0}\right]$ (13)

References

 Yang, L.; Hou, X.; and Xia, B. "A complete algorithm for automated discovering of a class of inequality-type theorems." Science China, F. Vol. 44, (2001): 33-49.