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I found that Internal`PolynomialFunctionQ performs much better than PolynomialQ.

Here is a huge random polynomial in 12 variables with around 120k terms:

myPoly = Product[(RandomInteger[{-2, 2}] + RandomInteger[{-2, 2}] a + 
  RandomInteger[{-6, 6}] b + RandomInteger[{-6, 6}] c + 
  RandomInteger[{-6, 6}] d + RandomInteger[{-6, 6}] e + 
  RandomInteger[{-2, 2}] f + RandomInteger[{-1, 1}] g + 
  RandomInteger[{-6, 6}] h + RandomInteger[{-6, 6}] i + 
  RandomInteger[{-6, 6}] j + RandomInteger[{-6, 6}] k + 
  RandomInteger[{-1, 1}] l), {go, 1, 8}] // Expand;

Let's make it not a polynomial in a by replacing the 1000th term with Sin[a]:

myPoly = ReplacePart[myPoly, 1000 -> Sin[a]];

So, now let's see how Internal`PolynomialFunctionQ and PolynomialQ perform:

AbsoluteTiming[Internal`PolynomialFunctionQ[myPoly, a]]
AbsoluteTiming[Internal`PolynomialFunctionQ[myPoly, b]]
AbsoluteTiming[Internal`PolynomialFunctionQ[myPoly, {a, b, c, d, e, f, g, h, i, j, k, l}]]
AbsoluteTiming[Internal`PolynomialFunctionQ[myPoly, {b, c, d, e, f, g, h, i, j, k, l}]]
(*  {0.033989, False}  *)
(*  {0.032627, True}   *)
(*  {0.056368, False}  *)
(*  {0.074603, True}   *)

and

AbsoluteTiming[PolynomialQ[myPoly, a]]
AbsoluteTiming[PolynomialQ[myPoly, b]]
AbsoluteTiming[PolynomialQ[myPoly, {a, b, c, d, e, f, g, h, i, j, k, l}]]
AbsoluteTiming[PolynomialQ[myPoly, {b, c, d, e, f, g, h, i, j, k, l}]]
(*  {3.98786, False}  *)
(*  {4.00939, True}   *)
(*  {3.222, False}  *)
(*  {3.32597, True}   *)

It seems like Internal`PolynomialFunctionQ performs 100 times better than PolynomialQ when checking for polynomialness in one variable, and about 50 times better for multiple variables.

Is anyone aware of this? Is Internal`PolynomialFunctionQ a low-level version of PolynomialQ?

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  • $\begingroup$ I doubt anyone can answer this, since it requires looking at the source code of the Wolfram Research Mathematica implementation of these algorithms, which only those who work at Wolfram are able to do. $\endgroup$ – Nasser Dec 27 '15 at 6:10
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First note they are not equivalent:

PolynomialQ[x + x[1], x]
Internal`PolynomialFunctionQ[x + x[1], x]
(*
  True
  False
*)

Second note that PolynomialQ does a lot of checking:

Trace[
 PolynomialQ[x^2 Sin[y], x],
 TraceInternal -> True]

Mathematica graphics

If you care to, you can verify that every factor of every term seems to be checked:

With[{e = (a + 2 b + 3 c + 4 y)^2 // Expand},
 Trace[
  PolynomialQ[e, x],
  TraceInternal -> True]
 ]

What exactly this checking consists of seems to be inaccessible. I've seen Integrate`FakeIntervalElement before, but I don't know what it's for or why it is used here.

On the OP's example, 600,000 expressions are checked, I suppose:

Count[myPoly, a | b | c | d | e | f | g | h | i | j | k | l, Infinity]
(*  602194  *)

Probably Internal`PolynomialFunctionQ is meant for a narrower range of use and goes straight to work on determining whether the variables appear only in nonnegative powers, etc.

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