# Polynomials with integer coefficients vs. polynomials with rational coefficients

My problem is very simple. From two different functions $f_1$ and $f_2$, I create two multivariate polynomials.

Because of theoretical reasons those two functions (evaluated with the same input $x$) should send the same output.

Now when I want to check this I always get false. The reason is the following I declare $P_1=f_1(x)$ and $P_2=f_2(x)$ and when I see the polynomials $P_1$ and $P_2$ they are actually equal but the first one has integer coefficients and the second one has rational coefficients (when the coefficient is $2$ in the first one, in the second the coefficient will be $2.$).

Here is the question, how can I formally check that I have found the same polynomial in both cases? I think this would involve to convert the coefficients of the first one into rational but I am not able to find in the help of Mathematica how to do it, and then to check equality with $P_1===P_2$.

• 2. is represented as floating point, not a Rational. IntegerQ[2.] is False; IntegerQ[2/1] is True (evaluates to 2). Jul 24, 2015 at 16:33
• Check documentation for Rationalize. Jul 24, 2015 at 16:53

 f1[x_, y_] = x + 2 y^2
f2[x_, y_] = x + 2. y^2

Quiet[{0} == Union @
Flatten @ Chop[
CoefficientList[f1[x, y], {x, y}] -
CoefficientList[f2[x, y], {x, y}]]]


True

The Quiet is needed to handle the case where the polynomials do not share all the same terms.

This will fail in the maybe pathological case where you have a numerically small high order term, e.g., 1 + x + 10.^-17 x^2 will show not equal to 1 + x.

You might also more simply do:

N[f1[x, y]] == N[f2[x, y]]