Corrigendum: “Could René Descrates have known this?”

ABSTRACT

Here we provide a correct version of Proposition 6 of [FKS]. No other results of the latter paper are affected.

KEYWORDS:

• real univariate polynomials
• Descartes rule of signs

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 12D99

This article refers to:

Could René Descartes Have Known This?

1. Correction

Paper [FKS] considers various aspects of the Descartes rule of signs for real univariate polynomials. In what follows, we always assume that all coefficients of such polynomials are nonzero. A sign pattern is a sequence of d + 1 plus or minus signs, $d\in \mathbb{N}$. For a given sign pattern σ with c sign changes and p sign preservations with c + p = d, its Descartes pair equals (c, p); for any k, $\ell \in \mathbb{N}\cup 0$, such that c − 2k ⩾ 0 and p − 2ℓ ⩾ 0, we call (c − 2k, p − 2ℓ) an admissible pair.

By the Descartes rule of signs, if the signs of the coefficients of a real polynomial P of degree d form the sign pattern σ, then P has c − 2k positive and p − 2ℓ negative roots for some k and ℓ as above. (P additionally has k + ℓ complex conjugate pairs of roots.)

Without loss of generality, we assume that the first symbol of each sign pattern is +. The sign pattern σ is called realizable with the admissible pair (c − 2k, p − 2ℓ) if there exists a monic degree d polynomial P whose signs of the coefficients are given by σ and which has exactly c − 2k positive and p − 2ℓ negative roots.

In paper [FKS], we consider (among others) sign patterns with exactly two sign changes (m pluses followed by n minuses followed by q pluses, m, n, $q\in \mathbb{N}$, m + n + q = d + 1). For such a sign pattern, all admissible pairs are of the form (0, v) or (2, v).

In the above notation, Part ii) of Proposition 6 in [FKS] should be corrected as follows:

The sign pattern σ is realizable with any pair of the form (2, v) except in the case when d and m are even and n = 1 (hence q is even).

In paper [FKS] the “except” part is missing, yet this exception follows from Proposition 4 of [FKS]. The above modification of the formulation of Proposition 6 does not affect the rest of the paper. In particular, for d = 8 the two sign patterns $$\begin{array}{c}\hfill (+,+,+,+,-,+,+,+,+)\mathrm{and}\\ \hfill (+,+,+,+,+,+,-,+,+)\end{array}$$are not realizable with the admissible pair (2, 0) as correctly stated in Theorem 10 of [FKS].

The realizability with admissible pairs of the form (2, v), v ⩾ 2, of the sign patterns with two sign changes is proved correctly, see the proof of Proposition 6 in [FKS]. Only the last paragraph of this proof (dealing with admissible pairs of the form (2, 1) for d odd or (2, 0) for d even) has to be changed as follows.

If d is even and the sign of at least one monomial x^s of odd degree is negative, then consider the polynomial U ≔ −x^s + ϵ(x^d + 1). For ϵ > 0 small enough, it has two positive simple and no other real roots. One can then perturb U by adding the remaining monomials with coefficients of the necessary signs and of moduli much smaller than ϵ to obtain a polynomial the signs of whose coefficients define the sign pattern σ. Upon dividing it by ϵ one obtains a monic polynomial with the sign pattern σ, with two positive simple and no other real roots.

Thus for d even, one can fail to realize a sign pattern σ with a given admissible pair only when there is a single minus sign, and it corresponds to a monomial of even degree. This is precisely the exceptional case.

If d is odd and σ contains a minus sign corresponding to a monomial of odd degree s, then consider the polynomial V ≔ x^d − x^s. It has a simple negative, a simple positive, an s-fold zero root, and no other real roots. Hence V + ϵ has two simple positive, one simple negative, and no other real roots for ϵ > 0 small enough. It remains to perturb V in order to have all monomials with nonzero coefficients as it was done for U.

If d is odd and σ contains a minus sign corresponding to a monomial of even degree h, then the polynomial W ≔ x^d − x^h has a simple positive, an h-fold zero and no other real roots. Hence, W + ϵ has a simple negative, two simple positive, and no other real roots, it remains to perturb W as it was done with U and V.

Acknowledgment

The necessity to modify Proposition 6 was pointed out to us by Y. Gati and H. Cheriha to whom we express our most sincere gratitude.