It is a bit strange to talk about the "roots" of a bivariate homogeneous polynomial. I guess that you mean the factors of the polynomial, not really the roots. Every bivariate homogeneous polynomial factors into (possibly complex) lines, so I guess that your question is how to check if your polynomial has multiple (linear) factors. In Computer Algebra, a polynomial with multiple factors is said to be "non-square-free", otherwise it is called "square-free". Calling your polynomial f(x,y), it is well-known that f(x,y) is square-free iff the greatest common divisor of f(x,y) and its partial derivatives with respect to x,y is constant. So compute that gcd (you can do that in your favourite computer algebra system, Maple, Mathematica, Sage, etc.), and see if you get a constant, or not.

Dear Mr. Juan Gerado Alcazar,

Thank you for your answer.

In fact, my question was how to determine if a bivariate homogeneous polynomial is square-free or not.

So if I understand well, I have to compute GCD(f(x,y),df(x,y)/dx) and GCD(f(x,y),df(x,y)/dy), and verify that we get a constant in the two cases in order to conclude that the polynomial is square-free ?

As a matter of fact, the object you're looking for is named the discriminant. Please take a look at http://pmpu.ru/vf4/detse/discrime

Section "Determinantal representation"

Dear Mr. Alexei Uteshev,

Thank you for your answer. In fact, it is exactly was I was looking for, and it joins the answer of Mr. Juan Gerado Alcazar.

Best regards.

Well, it is the GCD of the THREE polynomials (f, and both partial derivatives) that guarantees you squarefree-ness, in general. With the discriminant with respect to a variable, you check whether or not a polynomial and the derivative of the polynomial with respect to the same variable share a factor. But that does not necessarily imply that you have multiple factors. For instance, consider f(x,y)=xy. It is a bivariate, homegenous polynomial, and shares a factor (but a different one in each case!) with the partial derivative with respect to x, f_x=y, and with the partial derivative with respect to y, f_y=x. This would be detected by the vanishing of the two discriminants. But however, the polynomial does not have multiple factors. So I would recommend just to check if the GCD of the THREE polynomials f, f_x, f_y, is constant or not. In fact, for big polynomials, computing GCDs is cheaper than computing discriminants.

Dear Hamza,

If the polynomial P(x,y) is bivariate homogeneous of order n, then you can simply transform P into one variable polynomial of order n :

P(x,y) =yn p(x/y, 1) = yn f(u). ( u = x/y).

It helps to avoid the use of partial derivatives.

Then you can check f(u) for multiple roots (equivalent to various factors).

In this case: GCD and discriminants techniques are equivalent.

In fact, One can improve the techniques that are recommended by Juan and Alex to work for multivariate ( homogeneous or nonhomogeneous) polynomials too.

best wishes

Thank you all for your answers.

Yes I thinked about your solution Mr. Issam Kaddoure, but how about the case when there is a (0,0) solution, and may be a multiple root ?

Dear Hamza,

You are right. This is one of the trivial cases: (0,n) , (m,0) , (0,0) appeared as a multiple roots where P(x,y) has xr, ys or (xy)t as a common factor. We remove this term before proceeding the test.

Best regards