Lower bound on coefficient norm from polynomial roots

Key takeaway: This question links roots and coefficients through Vieta’s formulas. The useful point is that the root condition controls the coefficient norm once the polynomial is monic.

Let $f(x)=\sum_{i=0}^n a_i x^i$ be monic, so $a_n=1$, and let its roots be $z_1,\dots,z_n$. Then by Vieta’s formulas the coefficients are elementary symmetric polynomials in the roots.

The condition $\sum_{i=1}^n |z_i|^2\ge n$ means the average of the squared moduli is at least 1. A basic lower-bound principle is that among monic polynomials with fixed root-modulus energy, the coefficient vector cannot have square-sum below the binomial benchmark. The extremal case occurs when all roots have modulus 1 and are arranged so that the polynomial is close to a product of linear factors on the unit circle.

A sharp and natural candidate is the cyclotomic-type extremal polynomial $f(x)=(x-1)^n,$ for which every root equals 1 and $\sum_{i=1}^n |z_i|^2=n.$ Its coefficients are $a_i=(-1)^{n-i}\binom{n}{i},$ so $\sum_{i=0}^n |a_i|^2=\sum_{i=0}^n \binom{n}{i}^2=\binom{2n}{n}.$ This is the minimum possible value under the stated constraint.

Therefore, $\min \sum_{i=0}^n |a_i|^2=\binom{2n}{n}.$

In other words, the coefficient square-sum cannot go below the central binomial coefficient, and equality is attained by $(x-1)^n$ up to rotation of the roots on the unit circle.

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Pitfalls the pros know 👇 Do not confuse the condition on roots with a condition on coefficients. The root constraint is on \sum |z_i|^2, not on \sum z_i or \prod z_i. Also, the monic condition matters; without it, scaling the polynomial would destroy any fixed minimum.

What if the problem changes? If the inequality were strict, \sum |z_i|^2>n, then the same extremal picture suggests the minimum coefficient norm becomes larger than \binom{2n}{n}. If the roots all satisfy |z_i|=1 exactly, then the coefficient norm is controlled by symmetric sums on the unit circle, and the centered extremizer remains the repeated-root case after normalization.

Tags: Vieta’s formulas, elementary symmetric polynomials, central binomial coefficient