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We show that with high probability the number of real zeroes of a randompolynomial is bounded by the number of vertices on its Newton-Hadamard polygontimes the cube of the logarithm of the polynomial degree. A similar estimateholds for zeroes lying on any curve in the complex plane, which is the graph ofa Lipschitz function in polar coordinates. The proof is based on the classicalTur\'an lemma.

Real zeroes of random polynomials, I: Flip-invariance, Tur\'an's lemma, and the Newton-Hadamard polygon