# Mathematical notation for polynomial with given roots

I wonder if there is a mathematical notation for a polynomial defined by its roots. For example, a polynomial with $$N$$ roots $$z_m$$ ($$m=1,...,N)$$ can be written as $$\prod\limits_{m=1}^N{(z-z_m)}$$, but I am looking for something shorter like P$$^N(z_m)$$, indicating that it is a polynomial of order $$N$$ with the roots $$z_m$$ ($$m=1,...,N)$$.

• I don't know any specific notation, but I'd say that $\prod (z-z_m)$ is already kind of simple. Also (I know you're not saying this would be a good notation), choosing the notation $P^N(z_m)$ would be hard to read: the square of its third derivative, evaluated at $z_m$ would be written $((P^N(z_m))^{(3)}(z_m))^2$ (it is a bit dramatic, sorry for that). Commented Jul 10 at 10:04
• You are right, but the third derivative of $\prod\limits_{m=1}^N (z-z_m)$ would also get pretty long. Incidentally, I don't need any derivatives in my analysis, just products of polynomials.
– Mark
Commented Jul 11 at 6:42
• And you get to manipulate many of such polynomials? You could leave the exponent $N$ out, as it is clear that the degree of this polynomial is $N$. Also, you could write $P_{z_1,\dots,z_m}$ in order to be able to evaluate this polynomial without any confusion... Commented Jul 11 at 8:52