# Is there an approximate formula for the discriminant of a sparse polynomial?

Consider integer polynomials $$P \in \mathbb{Z}[X] \setminus \{0\}$$ of a degree $$D \geq 1$$ and without multiple complex roots. Let me introduce a notation $$d(P) := \frac{1}{D} \log{|\mathrm{Disc}(P)|}$$ for the logarithmic root discriminant of $$P$$. An outstanding open problem is to prove (or disprove) that $$d(P) \to \infty$$ under any sequence of polynomials with degrees $$D \to \infty$$. Indeed it seems rare to have $$d(P)$$ much smaller than $$\log{D}$$; though that can certainly happen, since for instance $$d(P) = \log{D} - \log{\log{D}} + \log{\log{\log{D}}} + O(1)$$ when $$P = \Phi_{p_1 \cdots p_k}$$ is a cyclotomic polynomial of a level equal the product of the first $$k$$ primes (asymptotically as $$k \to \infty$$).

Let $$\ell^1(P)$$ be the sum of the absolute values of the coefficients of $$P$$.

Question. Does $$d(P) = \log{D} + O(\log{\ell^1(P)}),$$ with an absolute implied coefficient?

This is really about a lower bound on the discriminant of a polynomial with small integer coefficients; the explicit upper bound $$d(P) \leq \log{D} + 2 \log{\ell^1(P)}$$ is certainly known to hold for all polynomials, thanks to Mahler's classical inequality. Besides this rudimentary observation we note that the answer is certainly positive for all trinomials, thanks to an explicit formula of Swan.

A part of this question is, in particular, whether all cyclotomic polynomials $$P = \Phi_N$$ have large enough coefficients to meet a discriminant lower bound $$d(P) - \log{D} \gg - \log{\ell^1(P)},$$ which seems not immediately obvious. (What is the asymptotic for the $$\ell^1$$ norm of $$\Phi_{p_1 \cdots p_k}$$, where $$2 = p_1 < p_2 < \cdots$$ is the sequence of primes?)