We identify the scaling region of a width O(n(-1)) in the vicinity of the accumulation points t = +/-1 of the real roots of a random Kac-like polynomial of large degree n. We argue that the density of the real roots in this region tends to a universal form shared by all polynomials with independent, identically distributed coefficients c;, as long as the second moment sigma = E(c(i)(2)) is finite. In particular, we reveal a gradual (in contrast to the previously reported abrupt) and quite nontrivial suppression of the number of real roots for coefficients with a nonzero mean value mun = E(c(i)) scaled as mu(n) similar to n(-1/2). [References: 28]
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