There is an undocumented function Internal`RadicalMemberQ which takes two arguments. But whatever input I try for the two arguments, it seems to crash the kernel (Mathematica 10.3 on Mac OS X). Can anyone guess what the function does based on the name of this function?

Internal`RadicalMemberQ[Sqrt[x], x]           (*Kernel crash*)

Internal`RadicalMemberQ[2, x]                 (*Kernel crash*)

Internal`RadicalMemberQ[a x^2 + 2 x - b, x]   (*Kernel crash*)
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    $\begingroup$ Internal`RadicalMemberQ["Saul Alinsky"]? $\endgroup$ – ciao Jun 12 '16 at 9:04
  • $\begingroup$ @ciao Maybe this can help. :) $\endgroup$ – yode Jun 12 '16 at 10:37
  • $\begingroup$ here are some uses that do not crash(10.1): Internal`RadicalMemberQ[Pi, Sqrt[2]] (*False*) Internal`RadicalMemberQ[2, Sqrt[2]] (*True*) $\endgroup$ – george2079 Jun 12 '16 at 17:00
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    $\begingroup$ @ciao So sorry I missed that comment last year. $\endgroup$ – Daniel Lichtblau Jul 30 '17 at 20:11

For an example,

Internal`RadicalMemberQ[x y, {x^2 y, x y^3}]

(* True *)

which means that $x y$ belongs to the radical of the polynomial ideal generated by $x^2 y$ and $x y^3$.

The method used to obtain the result is effectively Property 3 in the Gröbner Bases documentation, see http://reference.wolfram.com/language/tutorial/ComplexPolynomialSystems.html.

As is the case with many undocumented and internal functions, there is no obligation to avoid crashes or strange results when given unexpected input.

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