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RegularChains[ChainTools]

 IsInRadical
 test membership to the radical of a saturated ideal

 Calling Sequence IsInRadical(p, rc, R)

Parameters

 p - polynomial of R rc - regular chain of R R - polynomial ring

Description

 • The command IsInRadical(p, rc, R) returns true if and only if p belongs to the radical of the saturated ideal of rc.
 • This command is part of the RegularChains[ChainTools] package, so it can be used in the form IsInRadical(..) only after executing the command with(RegularChains[ChainTools]). However, it can always be accessed through the long form of the command by using RegularChains[ChainTools][IsInRadical](..).

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{ChainTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[y,x\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $\mathrm{sys}≔\left\{{x}^{2}+1,{\left(y+2x\right)}^{2}\right\}$
 ${\mathrm{sys}}{≔}\left\{{\left({y}{+}{2}{}{x}\right)}^{{2}}{,}{{x}}^{{2}}{+}{1}\right\}$ (2)

Note that this input system is already a regular chain.

 > $\mathrm{out}≔\mathrm{Triangularize}\left(\mathrm{sys},R\right);$$\mathrm{rc}≔{\mathrm{out}}_{1}$
 ${\mathrm{out}}{≔}\left[{\mathrm{regular_chain}}\right]$
 ${\mathrm{rc}}{≔}{\mathrm{regular_chain}}$ (3)
 > $\mathrm{Equations}\left(\mathrm{rc},R\right)$
 $\left[{{y}}^{{2}}{+}{4}{}{x}{}{y}{-}{4}{,}{{x}}^{{2}}{+}{1}\right]$ (4)
 > $\mathrm{NumberOfSolutions}\left(\mathrm{rc},R\right)$
 ${4}$ (5)

Is $y+2x$ in the saturated ideal of rc?

 > $\mathrm{IsInSaturate}\left(y+2x,\mathrm{rc},R\right)$
 ${\mathrm{false}}$ (6)

Is $y+2x$ is the radical of the saturated ideal of rc?

 > $\mathrm{IsInRadical}\left(y+2x,\mathrm{rc},R\right)$
 ${\mathrm{true}}$ (7)

The function Triangularize can remove the squares as follows.

 > $\mathrm{out}≔\mathrm{Triangularize}\left(\mathrm{sys},R,\mathrm{radical}=\mathrm{yes}\right);$$\mathrm{sfrc}≔{\mathrm{out}}_{1}$
 ${\mathrm{out}}{≔}\left[{\mathrm{regular_chain}}\right]$
 ${\mathrm{sfrc}}{≔}{\mathrm{regular_chain}}$ (8)
 > $\mathrm{Equations}\left(\mathrm{sfrc},R\right);$$\mathrm{NumberOfSolutions}\left(\mathrm{sfrc},R\right)$
 $\left[{y}{+}{2}{}{x}{,}{{x}}^{{2}}{+}{1}\right]$
 ${2}$ (9)

Is $y+2x$ in the saturated ideal of sfrc?

 > $\mathrm{IsInSaturate}\left(y+2x,\mathrm{sfrc},R\right)$
 ${\mathrm{true}}$ (10)

 See Also

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