# Real univariate polynomials with given signs of coefficients and simple real roots

• V. P. Kostov Université Côte d'Azur, CNRS, LJAD, France
Keywords: standard discriminant, Descartes’ rule of signs, contractibility

### Abstract

We continue the study of different aspects of Descartes' rule of signs and discuss the connectedness of the sets of real degree $d$ univariate monic polynomials (i.~e. with leading coefficient $1$) with given numbers $\ell ^+$ and $\ell ^-$ of positive and negative real roots and given signs of the coefficients; the real roots are supposed all simple and the coefficients all non-vanishing.
That is, we consider the space $\mathcal{P}^d:=\{ P:=x^d+a_1x^{d-1}+\dots +a_d\}$, $a_j\in \mathbb{R}^*=\mathbb{R}\setminus \{ 0\}$, the corresponding sign patterns $\sigma=(\sigma_1,\sigma_2,\dots, \sigma_d)$, where $\sigma_j=$sign$(a_j)$, and the sets $\mathcal{P}^d_{\sigma ,(\ell ^+,\ell ^-)}\subset \mathcal{P}^d$ of polynomials with given triples $(\sigma ,(\ell ^+,\ell ^-))$.
We prove that for degree $d\leq 5$, all such sets are connected or empty. Most of the connected sets are contractible, i.~e. able to be reduced to one of their points by continuous deformation.
Empty are exactly the sets with $d=4$, $\sigma =(-,-,-,+)$, $\ell^+=0$, $\ell ^-=2$, with $d=5$, $\sigma =(-,-,-,-,+)$, $\ell^+=0$, $\ell ^-=3$, and the ones obtained from them under the $\mathbb{Z}_2\times \mathbb{Z}_2$-action
defined on the set of degree $d$ monic polynomials by its two generators which are two commuting  involutions: $i_m\colon P(x)\mapsto (-1)^dP(-x)$ and $i_r\colon P(x)\mapsto x^dP(1/x)/P(0)$.

We show that for arbitrary $d$, two following sets are contractible:

1) the set of degree $d$ real monic polynomials having all coefficients positive and with exactly $n$ complex  conjugate pairs of roots ($2n\leq d$);

2) for $1\leq s\leq d$, the set of real degree $d$ monic polynomials with exactly $n$ conjugate pairs ($2n\leq d$) whose first $s$ coefficients are positive and the next $d+1-s$ ones are negative.

For any degree $d\geq 6$, we give an example of a set $\mathcal{P}^d_{\sigma ,(\ell^+,\ell^-)}$  having $\Lambda (d)$ connected compo\-nents, where $\Lambda (d)\rightarrow \infty$ as $d\rightarrow \infty$.

### Author Biography

V. P. Kostov, Université Côte d'Azur, CNRS, LJAD, France

Université Côte d'Azur, CNRS, LJAD, France

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Published
2024-03-19
How to Cite
Kostov, V. P. (2024). Real univariate polynomials with given signs of coefficients and simple real roots. Matematychni Studii, 61(1), 22-34. https://doi.org/10.30970/ms.61.1.22-34
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