Roots of a random polynomial!

Let P(x)P(x) be a polynomial of degree dd, given by P(x)=c0+c1x+c2x2++cdxd.P(x)=c_0+c_1x+c_2x^2 +\ldots + c_dx^d. Each of the coefficients are chosen independently and uniformly at random from the pp-element set Fp={0,1,2,,p1},\mathbb{F}_p=\{0,1,2, \ldots, p-1\}, where pp is a given prime number.

An element kFpk \in \mathbb{F}_p is said to be a root of the polynomial modulo pp if and only if P(k)0(modp)P(k)\equiv 0 \pmod p (i.e. the prime pp divides the integer P(k)P(k)).

Since the coefficients are random, so is the polynomial P(x)P(x) and so are its roots (defined strictly in the above sense). For p=2017p=2017 and d=1008d=1008, find the expected number of roots of such a random polynomial.

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