# Universal properties of residual moments in heavy-fermion metals

Ewan Scott  Department of Mathematics, University College London, Gordon St.,
London WC1H 0AY, United Kingdom  Michal Kwasigroch  Department of Mathematics,
University College London, Gordon St., London WC1H 0AY, United Kingdom
Trinity College, Cambridge, CB2 1TQ, United Kingdom

###### Abstract

We identify common features, seen across a wide range of actinide and
lanthanide based heavy-fermion metals, that we associate with the universal
properties of the residual moments generated by Kondo hybridisation. The
features include the opposing anisotropies of the effective moments and Curie-
Weiss constants, the reorientation of the magnetic easy axis, the hard-axis
susceptibility maximum and the hard-axis metamagnetic transition. The
extension of large-$N$ Read-Newns theory to the underscreened Kondo lattice
allows us to study large magnetic fluctuations of residual local moments in
the presence of strong $f$-$d$ hybridisation. We derive an effective
Hamiltonian for the residual moments and show that competing single-ion and
exchange anisotropies are responsible for the easy-axis reorientation. We
identify a strong-coupling regime, where the temperature of the easy-axis
reorientation is directly proportional to the coherence temperature and find
good agreement across several Eu and U based heavy-fermion families, as well
as a weak-coupling regime, where the easy axis flips just before the onset of
magnetic order, as is seen in many Ce and Yb heavy-fermion compounds. We show
that the exchange anisotropy of the residual moments can also generate a
maximum in the hard-axis susceptibility, and find good agreement with UTe2 at
low pressures, suggesting that this material is particularly well described by
the single-channel $S=1$ model. Finally, we speculate on the possible pairing
interactions generated by the fluctuations of the residual moments.

††preprint: APS/123-QED

Introduction. – Heavy-fermion systems entwine electronic and magnetic degrees
of freedom leading to a rich variety of phenomena such as high field
superconductivity mediated by spin fluctuations Knafo  _et al._ (2021); Lewin
_et al._ (2023); Aoki  _et al._ (2022, 2019), metamagnetic transitions Miyake
_et al._ (2019), or the reorientation of the easy axis Hafner  _et al._
(2019). The interweaving begins with the formation of the flat heavy-fermion
bands as the scattering of electrons from local magnetic moments becomes
coherent. The emergent Fermi liquid excitations coexist with the residual
local moments that form. Although many general questions remain, this ’two-
fluid’ picture is a useful perspective in which to view a number of
experimental observations Yang  _et al._ (2008); Lonzarich  _et al._ (2016a,
b); Ji  _et al._ (2022), as well as theoretical results Gleis  _et al._
(2023); Raczkowski and Assaad (2020); Perkins  _et al._ (2007a), where Kondo
hybridisation has been found to coexist with magnetic order. The emergent
residual moments and quasiparticles originate from the same microscopic
building blocks and therefore remain entwined. For example, anisotropy of the
moments, that was initially of single-ion character, becomes non-local after
heavy-fermion band formation, generating anisotropic magnetic fluctuations,
and potentially, superconducting pairing.

The rich phenomenology of local-moment magnetism in heavy-fermion metals is
generally sensitive to the underlying crystal structure, anisotropy, or
valency of the local-moment ions. Nonetheless, across a wide range of
lanthanide and actinide compounds some robust universal features emerge Hafner
_et al._ (2019); Aoki  _et al._ (2019); Ōnuki  _et al._ (2020). One example is
the reorientation of the magnetic easy axis at low temperatures, that is
strongly correlated with the presence of the coherence maximum in the
material’s resistivity, as well as opposing anisotropies in the effective
magnetic moments and Curie-Weiss constants.

Figure 1: Top: The easy-axis reorientation temperature, $T_{\rm cross}$,
against the coherence temperature (from the coherence maximum in the
resistivity), $T^{*}_{\rm coh}$, for three compounds: UTe2 Li  _et al._
(2021); Ajeesh  _et al._ (2024), UCo1-xIrxGe Hovančík  _et al._ (2022); R.
Troć and Samsel-Czekała (2010), and EuCu2(Ge1-xSix)2 Iha  _et al._ (2018). The
direct proportionality fits suggest that the same energy scale is responsible
for the development of coherence and easy-axis reorientation. (For
UCo1-xIrxGe, the fit does not include the stars, and for EuCu2(Ge1-xSix)2 it
does not include the highest $T_{\rm cross}$, which is a tentative
prediction.) Bottom: A plot of the a-axis resistivity and modified Curie-Weiss
fits of b and c axis susceptibilities for UCo0.98Ir0.02Ge showing $T^{*}_{\rm
coh}$ and $T_{\rm cross}$. A detailed description of all the plotted data and
any extrapolations used can be found in Table I.

The reorientation of the easy axis is identified by a crossing of magnetic
susceptibilities for fields along two perpendicular directions. This trend was
first noticed in Ref. Hafner  _et al._ (2019) across a wide range of Yb and Ce
based Kondo ferromagnets, where the easy axis flips by 90 degrees just before
the onset of magnetic order. Here, we widen the universality of this
phenomenon by identifying Eu and U based heavy-fermion magnetic compounds,
with multiple $f$-electrons per magnetic ion, that undergo easy-axis
reorientation Li  _et al._ (2021); Hovančík  _et al._ (2022); Pospíšil  _et
al._ (2017); Iha  _et al._ (2018); Szlawska  _et al._ (2023); Bleckmann  _et
al._ (2010); Valenta  _et al._ (2018); Hiranaka  _et al._ (2013).

The universal features are not limited to the reorientation of the easy axis.
In U and Eu based magnets we notice that the reorientation is often linked to
the presence of a maximum in the temperature-dependence of the susceptibility
along the axis which loses its ’easy-axis status’ upon cooling Li  _et al._
(2021); Hovančík  _et al._ (2022); Pospíšil  _et al._ (2017); Iha  _et al._
(2018); Szlawska  _et al._ (2023); Hiranaka  _et al._ (2013). This hard-axis
maximum is in turn connected to a metamagnetic transition when a high magnetic
field is applied along the axis at lower temperatures Li  _et al._ (2021);
Hovančík  _et al._ (2022); Pospíšil  _et al._ (2017); Hiranaka  _et al._
(2013). In fact, looking at all the compounds that we have surveyed, it
appears that a metamagnetic transition in a high field applied along the hard
axis implies a maximum in the the susceptibility along this axis at a higher
temperature, which in turn implies that it becomes the easy axis at an even
higher temperature.

Unlike in Ce or Yb based materials, in Eu or U based heavy-fermion compounds
the reorientation of the easy axis can take place far away from the onset of
magnetic order, or even in its absence. For example, UCoGe has a coherence
maximum in the resistivity response along the $b$-axis at $T^{\ast}_{\rm
coh}=170$ K, a Curie temperature of $T_{C}=2.5$ K and a projected temperature
of easy axis reorientation of $T_{\rm cross}=427$ K, obtained from modified
Curie-Weiss (mCW) fits 111A mCW fit simply involves adding a constant offset
to the usual Curie-Weiss susceptibility. for measurements up to $400$ K R.
Troć and Samsel-Czekała (2010). (See the bottom panel of Fig. 1 for an example
of $T_{\rm cross}$ obtained from mCW fits and $T_{\rm coh}^{*}$ from the
coherence maximum in the resistivity.) Remarkably, we find that, in these
compounds, the crossing and coherence temperatures appear to be directly
proportional, showing that the same energy scale is responsible for both (see
Fig. 1 (top), with a detailed description of all the plotted data given in
Table I). The formation of the heavy fermion bands, which is responsible for
$T^{\ast}_{\rm coh}$ and, we believe, for $T_{\rm cross}$, takes place at a
temperature much higher than the onset of magnetic order pointing to the
underscreened nature of the underlying Kondo lattice and a regime where the
Kondo energy, $T_{K}$, is much larger than the RKKY energy scale, $T_{\rm
RKKY}$, that is not captured by the usual Doniach argument Read  _et al._
(1984).

Motivated by the above observation, we present a generalisation of the
standard large-$N$ Read-Newns theory to the underscreened Kondo lattice, that
will allow us to study heavy-fermion magnetism in a regime, where $T_{\rm
RKKY}\ll T_{K}$. We present here the main results of our approach as well as
experimental fits with the detailed derivations published elsewhere Scott and
Kwasigroch . We focus on the underscreened $S=1$ Kondo lattice with a single-
ion anisotropy, as it already captures a wide range of phenomena such as easy-
axis reorientation, competing Curie-Weiss and effective moment anisotropies,
and the maximum in the low temperature hard-axis susceptibility.

Model. – The underscreened $S=1$ Kondo lattice with a generic single-ion
anisotropy is described by the following Hamiltonian

$\displaystyle H=$ $\displaystyle
J_{K}\sum_{i}\mathbf{S}(\mathbf{r}_{i})\cdot\mathbf{s}(\mathbf{r}_{i})+\sum_{\mathbf{k}\sigma}\epsilon_{\mathbf{k}}c^{\dagger}_{\sigma}(\mathbf{k})c_{\sigma}(\mathbf{k})$
(1) $\displaystyle+\sum_{\eta
i}D^{\eta}\left(S^{\eta}(\mathbf{r}_{i})\right)^{2},$

where $J_{K}>0$ is the Kondo coupling between local spin $\mathbf{S}_{i}$ and
the conduction electron spin $\mathbf{s}_{i}$. For simplicity, we have assumed
that electrons and spins occupy the same Bravais lattice sites.
$\epsilon_{\mathbf{k}}$ is the linearised conduction band in the vicinity of
the Fermi surface such that $|\epsilon_{\mathbf{k}}|<\Lambda$. $D^{x,y,z}$ is
a general single-ion anisotropy for spin-1 222Of course one of the
$D^{\gamma}$ can always be set to 0.. We will work away from the insulating
limit, i.e., in the heavy Fermi liquid phase.

We write the $S=1$ local moments in the Schwinger fermion representation
Perkins  _et al._ (2007b); Ko  _et al._ (2013),

$S^{\eta}(\mathbf{r}_{i})=\sum_{a=1,2}S^{\eta}_{a}(\mathbf{r}_{i})=\frac{1}{2}\sum_{a\sigma\sigma^{\prime}}f_{a\sigma}^{\dagger}(\mathbf{r}_{i})\sigma^{\eta}_{\sigma\sigma^{\prime}}f_{a\sigma^{\prime}}(\mathbf{r}_{i}),$
(2)

where $a=1,2$ index the two spin-1/2 fermions, $\sigma,\sigma^{\prime}$ denote
the up/down spin indices, $\sigma^{x,y,z}$ are Pauli matrices and
$f_{a\sigma}(\mathbf{r}_{i})$ obey the usual fermionic anticommutation
relations. We will also promote the $SU(2)$ spin degrees of freedom to
$SU(N)$. This can be done in a number of ways and we will choose to have $N/2$
replicas of spin-up states and an equal number of spin-down states on each
lattice site, with half of them occupied, so that we are connected most
straightforwardly to the physical $N=2$ limit. $N$ will be the control
parameter for the large-$N$ theory. The Hamiltonian can be written (up to a
constant) as

$\displaystyle H_{N}=$
$\displaystyle\frac{J_{K}}{N}\sum_{a\alpha\sigma\beta\sigma^{\prime}i}f^{\dagger}_{a\alpha\sigma}(\mathbf{r}_{i})f_{a\beta\sigma^{\prime}}(\mathbf{r}_{i})c^{\dagger}_{\beta\sigma^{\prime}}(\mathbf{r}_{i})c_{\alpha\sigma}(\mathbf{r}_{i})$
$\displaystyle+\sum_{\mathbf{k}\alpha}\epsilon_{\mathbf{k}}c^{\dagger}_{\alpha\sigma}(\mathbf{k})c_{\alpha\sigma}(\mathbf{k})+\sum_{i,\alpha}D^{\eta}\left(\sum_{a}S^{\eta}_{a\alpha}(\mathbf{r}_{i})\right)^{2},$

where

$\displaystyle
S^{\eta}_{a\alpha}(\mathbf{r}_{i})=\sum_{\sigma\sigma^{\prime}}f^{\dagger}_{a\alpha\sigma}(\mathbf{r}_{i})\sigma^{\eta}_{\sigma\sigma^{\prime}}f_{a\alpha\sigma^{\prime}}(\mathbf{r}_{i}),$
(3)

and $\alpha,\beta$ index the spin up/down replicas. We have rescaled the Kondo
coupling $J_{K}\rightarrow(2J_{K}/N)$ so that energy density remains finite as
$N\rightarrow\infty$, and $H_{N=2}=H$. The enlargement of the Hilbert space
through the Schwinger fermion representation comes with several constraints
and associated gauge symmetries:

$\displaystyle\hat{n}^{f}_{i}$ $\displaystyle:=$
$\displaystyle\sum_{a}\hat{n}^{f}_{ai}=\sum_{a\alpha\sigma}f^{\dagger}_{a\alpha\sigma}(\mathbf{r}_{i})f_{a\alpha\sigma}(\mathbf{r}_{i})=N,$
$\displaystyle\hat{{T}}^{\eta}_{i}$ $\displaystyle:=$
$\displaystyle\frac{1}{2}\sum_{aa^{\prime}\alpha\sigma}f^{\dagger}_{a\alpha\sigma}(\mathbf{r}_{i})\tau^{\eta}_{aa^{\prime}}f_{a^{\prime}\alpha\sigma}(\mathbf{r}_{i})=0,$
(4)

where $\tau^{x,y,z}$ are again Pauli matrices, which now act on the
2-dimensional $a$ subspace. We can understand the constraints
$\hat{n}^{f}_{i}$ and $\hat{T}^{z}_{i}$ as together ensuring that there are
$\hat{n}^{f}_{1i}=\frac{N}{2}$ $f_{1}$-fermions and
$\hat{n}^{f}_{2i}=\frac{N}{2}$ $f_{2}$-fermions on each lattice site. The
constraints $\hat{T}^{x,y}$ couple the $SU(N)$ $f_{1}$ and $f_{2}$ spins,
which for $N=2$ projects out the singlet state and corresponds to an infinite
Hund’s coupling between $f_{1}$ and $f_{2}$ moments. The constraints generate
the $U(2)\sim U(1)\times SU(2)$ group of local gauge transformations.

The isotropic ($D^{\eta}=0$) system is controlled by the fixed-gauge saddle
point as $N\rightarrow\infty$. Read-Newns large-$N$ theory introduces the
hybridisation field

$\begin{pmatrix}\hat{V}_{1i}\\\
\hat{V}_{2i}\end{pmatrix}=\frac{1}{N}\sum_{\alpha\sigma}\begin{pmatrix}f^{\dagger}_{1\alpha\sigma}(\mathbf{r}_{i})c_{\alpha\sigma}(\mathbf{r}_{i})\\\
f^{\dagger}_{2\alpha\sigma}(\mathbf{r}_{i})c_{\alpha\sigma}(\mathbf{r}_{i})\end{pmatrix},$
(5)

and the $U(2)$ gauge symmetry can be used to set $\hat{V}_{2i}=0$ and
$\hat{V}_{1i}>0$. We emphasise that this is not done just on the level of
expectation values. Quantum fluctuations (in imaginary time) of $\hat{V}_{2i}$
are also quenched. $f_{2}$-fermions are thus decoupled from the conduction
electrons in the hybridisation channel. The constraint-enforcing gauge fields
develop a non-zero stiffness softening the constraints on $\hat{n}^{f}_{1i}$
and $\hat{T}^{\eta}_{i}$. A residual $U(1)$ gauge symmetry remains so that the
constraint on $\hat{n}^{f}_{2i}$ is still enforced exactly.

The fluctuations of the constraint-enforcing gauge fields around the saddle
point generate the following interactions

$\displaystyle\Delta H_{\rm
fluc.}=G_{\lambda\lambda}(0)\sum_{i}\left(\hat{n}^{f}_{1i}-\frac{N}{2}\right)^{2}+$
$\displaystyle\;+G_{\phi^{\dagger}\phi}(0)\sum_{i,\alpha\sigma\beta\sigma^{\prime}}f_{1\alpha\sigma}^{\dagger}(\mathbf{r}_{i})f_{1\beta\sigma^{\prime}}(\mathbf{r}_{i})f_{2\beta\sigma^{\prime}}^{\dagger}(\mathbf{r}_{i})f_{2\alpha\sigma}(\mathbf{r}_{i}),$

where the gauge-field Green’s functions in the low-energy limit scale as
$G_{\lambda\lambda}(0)\sim T_{K}/N$ and $G_{\phi^{\dagger}\phi}(0)\sim
T_{K}(N\log(T/T_{K}))^{-1}<0$ 333We note that the residual Hund-like
interaction does not decay logarithmically but tends to a constant in the
insulating half-filling limit.. The infinite Hubbard-like interaction that
enforced the constraint on $\hat{n}^{f}_{1i}$ has been renormalised to a
finite interaction and so has the infinite Hund-like interaction that enforced
the constraint on $\hat{T}^{x,y}_{i}$. The latter is the residual
ferromagnetic interaction between the unscreened $f_{2}$-moments and the heavy
fermions ($f_{1}$-fermions are now part of the heavy fermion quasiparticle).
The stiffness of the gauge-field whose fluctuations generate the Hund-like
interaction diverges logarithmically in the zero-temperature limit leading to
logarithmically decaying residual ferromagnetic interaction. This agrees with
the results for a single impurity Nozières, Ph. and Blandin, A. (1980);
Allerdt  _et al._ (2018).

While it is true that fractional violations of the constraints
$\frac{1}{N}\langle(\hat{T}^{\eta})^{2}\rangle^{1/2}\sim 1/\sqrt{N}$ are small
for large-$N$, it is remarkable that the $N=\infty$ saddle point can capture
the behaviour of physical systems with $N=2$, and that it has even provided
quantitative agreement in many cases Coleman and Andrei (1986). Intuitively,
the system flows to the large-$N$ fixed point in the RG sense. The long-range
coherence of the hybridisation field allows for the averaging over close-by
(in imaginary time and real space) hybridisation fields, and effectively a
local hybridisation process is coupled to the average of many surrounding
ones. In many ways, this is what the introduction of additional fictitious
spin flavours is equivalent to.

Residual local moments. – The unhybridised $f_{2}$-moments thus decouple in
the zero-temperature limit as the residual ferromagnetic interaction decays
logarithmically to zero. This would be the complete story for a single
impurity, but for a Kondo lattice, the macroscopic degeneracy of the
$f_{2}$-moment subspace can be broken by a coupling that leaves a vanishing
imprint on the $f_{1}$-$c$ heavy fermion subspace. Because of the $U(2)$ gauge
symmetry, the $f_{2}$-fermions can be decoupled from the conduction electrons
in the hybridisation channel but not the magnetic channel since magnetisation,
$S^{\eta}(\mathbf{r}_{i})$, is a gauge-invariant physical operator.
Unfortunately, the fixed-gauge saddle point expansion (up to first order in
$1/N$) does not include interactions in this channel and the Kondo-hybridised
phase is stable against RKKY magnetism. However, the fact that for any finite
$N$, as $J_{K}\rho\rightarrow 0$, the conventional RKKY ground state becomes
favourable over the Kondo-hybridised phase can be picked up variationally and
this is indeed the original Doniach argument Doniach (1977).

Motivated by the above observation, we consider the following variational
ansatz for the state of the system at $T\ll T_{K}$

$\displaystyle|\Psi\rangle=\prod_{\mathbf{k\alpha\sigma\nu}}\left(u^{\nu}_{\mathbf{k}\alpha\sigma}c^{\dagger}_{\mathbf{k}\alpha\sigma}+v^{\nu}_{\mathbf{k}\alpha\sigma}f^{\dagger}_{1\mathbf{k}\alpha\sigma}\right)|0\rangle\otimes\prod_{i}|\psi_{2i}\rangle,$
(7)

where $\nu$ indexes two heavy fermion bands and $|\psi_{2i}\rangle$ is the
state of the $i$-th $f_{2}$-moment. By employing the usual static
approximation for the $f_{2}$-moments, calculating
$\langle\Psi|H_{N}|\Psi\rangle$ with
$\\{u^{\nu}_{\mathbf{k}\alpha\sigma},v^{\nu}_{\mathbf{k}\alpha\sigma}\\}$
chosen to minimise the expectation value, we can derive the effective RKKY
interaction between $f_{2}$-moments that is mediated by the heavy fermions.
Expanding to quadratic order in $1/N$ and $D^{\eta}$ and linear order in an
external magnetic field $h$, we obtain

$\displaystyle H_{\rm
eff}\\!=\\!-\frac{1}{2}\sum_{\mathbf{q}\alpha\eta}\mathcal{J}^{\eta}(\mathbf{q})S^{\eta}_{2\alpha}(\mathbf{q})S^{\eta}_{2\alpha}(-\mathbf{q})\\!-\\!\sqrt{N_{s}}\sum_{\eta\alpha}\tilde{h}^{\eta}S^{\eta}_{2\alpha}(\mathbf{0}),$

where

$\displaystyle\mathcal{J}^{\eta}(\mathbf{q})=\frac{4J_{K}^{2}}{N^{2}}\chi_{ee}(\mathbf{q})+\frac{8J_{K}}{N}D^{\eta}\chi_{fe}(\mathbf{q})+(2D^{\eta})^{2}\chi_{ff}(\mathbf{q}),$
$\displaystyle\frac{\tilde{h}^{\eta}}{h}\\!=\\!1\\!-\\!\frac{2J_{K}}{N}(\chi_{ee}(\mathbf{0})+\chi_{fe}(\mathbf{0}))\\!-\\!2D^{\eta}\left(\chi_{ff}(\mathbf{0})+\chi_{fe}(\mathbf{0})\right).$
(9)

The susceptibilities of the spin-$1/2$ Kondo lattice $N=\infty$ ground state
are given by

$\displaystyle\chi_{ee}(\mathbf{q})$ $\displaystyle=$
$\displaystyle\frac{1}{2N_{s}}\sum_{\mathbf{k},\nu\nu^{\prime}}|u^{\nu}_{\mathbf{k}}|^{2}|u^{\nu^{\prime}}_{\mathbf{k}+\mathbf{q}}|\frac{n(E^{\nu^{\prime}}_{\mathbf{k}+\mathbf{q}})-n(E^{\nu}_{\mathbf{k}})}{E^{\nu}_{\mathbf{k}}-E^{\nu^{\prime}}_{\mathbf{k}+\mathbf{q}}},$
$\displaystyle\chi_{fe}(\mathbf{q})$ $\displaystyle=$
$\displaystyle\frac{1}{2N_{s}}\sum_{\mathbf{k},\nu\nu^{\prime}}\bar{v}^{\nu}_{\mathbf{k}}v^{\nu^{\prime}}_{\mathbf{k}+\mathbf{q}}u^{\nu}_{\mathbf{k}}\bar{u}^{\nu^{\prime}}_{\mathbf{k}+\mathbf{q}}\frac{n(E^{\nu^{\prime}}_{\mathbf{k}+\mathbf{q}})-n(E^{\nu}_{\mathbf{k}})}{E^{\nu}_{\mathbf{k}}-E^{\nu^{\prime}}_{\mathbf{k}+\mathbf{q}}},$
$\displaystyle\chi_{ff}(\mathbf{q})$ $\displaystyle=$
$\displaystyle\frac{1}{2N_{s}}\sum_{\mathbf{k},\nu\nu^{\prime}}|v^{\nu}_{\mathbf{k}}|^{2}|v^{\nu^{\prime}}_{\mathbf{k}+\mathbf{q}}|\frac{n(E^{\nu^{\prime}}_{\mathbf{k}+\mathbf{q}})-n(E^{\nu}_{\mathbf{k}})}{E^{\nu}_{\mathbf{k}}-E^{\nu^{\prime}}_{\mathbf{k}+\mathbf{q}}},$
(10)

where $\chi_{ee}(\mathbf{q})$ ($\chi_{fe}(\mathbf{q})$) gives the magnetic
susceptibility of the conduction electrons (spin-1/2 local moments) when a
$\mathbf{q}$-modulated magnetic field is applied isothermally to conduction
electrons only. Similarly, $\chi_{ff}(\mathbf{q})$ gives the magnetic
susceptibility of the spin-1/2 local moments, when the applied field only
couples to them.

The effective Hamiltonian in Eq. Universal properties of residual moments in
heavy-fermion metals is the main result of our work and in the isotropic and
insulating limit agrees with the Hamiltonian derived in Ref. Ko  _et al._
(2013) via a slightly different approach. There, the difference between
$H_{N=2}$ and mean-field Hamiltonian (exact for $N=\infty$) was considered as
the residual perturbative interaction and the effective Hamiltonian for the
$f_{2}$-subspace was obtained. They showed that the results of this approach
agreed well with DMRG calculations for a 1D Kondo lattice problem.

Figure 2: The ratios $T_{\rm N/C}/T_{\rm cross}$ and $T_{\rm cross}/T^{*}_{\rm
coh}$ plotted for several compounds. Yb and Ce compounds Hafner  _et al._
(2019); Krellner  _et al._ (2011); Steppke  _et al._ (2013); Krellner and
Geibel (2008); Krellner  _et al._ (2007, 2008); Lausberg  _et al._ (2013);
Klingner  _et al._ (2011); Gruner  _et al._ (2012); Rai  _et al._ (2019) are
in the weak coupling regime and UCo1-xIrxGe transitions between regimes with
$x$. Inset: the ratios $T_{\rm RKKY}/T_{\rm cross}$ and $(N/2)T_{\rm
cross}/T_{K}$ plotted against the Kondo coupling strength, $J_{K}\rho$. There
are two regimes: Weak coupling, $T_{\rm RKKY}/T_{\rm cross}\sim 1$, and strong
coupling, $(N/2)T_{\rm cross}/T_{K}>0$. The divergence of the red curve
signals a return to the conventional RKKY phase when $T_{\rm RKKY}\gtrsim
T_{K}$.

Easy-axis reorientation. – Since $\chi(\mathbf{0})$ are all positive Scott and
Kwasigroch , the anisotropy of the effective field, $\tilde{h}^{\eta}$, and
the anisotropy of the ferromagnetic exchange $\mathcal{J}(\mathbf{0})$ are of
opposite sign to first order in $D^{\eta}$. Hard-axis anisotropy $D^{\eta}>0$
leads to a reduction of the residual effective moment but a reinforcement of
RKKY exchange between the residual moments along that direction. In
particular, a ground state with FM order would have moments along the
direction with the the highest $D^{\eta}$, and hence greatest ferromagnetic
exchange but smallest susceptibility at high temperature in the free-moment
regime – this is the hard-direction ordering discussed in Ref. Scott and
Kwasigroch ; Kwasigroch  _et al._ (2022). The single-ion and exchange
anisotropies respectively correspond to the anisotropy in the effective
moments $\mu_{\rm eff}^{\eta}$ and Curie-Weiss constants $\Theta_{\rm
CW}^{\eta}$. Indeed, in agreement with $H_{\rm eff}$, many of the compounds
surveyed have $(\Delta\mu_{\rm eff}^{\eta})/\mu_{\rm
eff}\sim-(\Delta\Theta_{\rm eff}^{\eta})/T_{\rm cross}$, where $\Delta$ takes
the difference between the two directions involved in the easy-axis
reorientation and $\mu_{\rm eff}$ is the average effective moment for these
two directions. The competition between the two anisotropies leads to a
reorientation of the easy axis at a temperature of

$\displaystyle T_{\rm
cross}=\frac{2J_{K}}{N}\frac{\chi_{fe}(\mathbf{0})}{\chi_{ff}(\mathbf{0})+\chi_{fe}(\mathbf{0})}$
$\displaystyle+\frac{J_{K}^{2}}{N^{2}}\left(\chi_{ee}(\mathbf{0})-\frac{\chi_{fe}(\mathbf{0})(\chi_{ee}(\mathbf{0})+\chi_{fe}(\mathbf{0}))}{\chi_{ff}(\mathbf{0})+\chi_{fe}(\mathbf{0})}\right),$
(11)

to quadratic order in $1/N$ and first non-vanishing order in $D^{\eta}$. We
identify two limits as depicted in Fig. 2. In the strong-coupling limit
$J_{K}\rho\gg 1$ (this corresponds to $NJ\rho\gg 1$), we have to first non-
vanishing order in $D^{\eta}$ and $1/N$

$\displaystyle T_{\rm cross}=\frac{J_{K}}{4N}=\frac{T_{K}}{N},$ (12)

where $T_{K}$ is Kondo energy scale 444We take it to be one half of the
difference between the energy of the $N=\infty$ ground state and the
$N=\infty,J=0$ Fermi gas state. For $J\rho\gg 1$, $T_{K}$ is also precisely
the temperature at which the hybridisation field condenses for $N=\infty$ and
half-filling.. In the weak-coupling limit, for any fixed $N$, as
$J_{K}\rho\rightarrow 0$, the crossing temperature tends to $T_{\rm
RKKY}=(J_{K}/N)^{2}\chi_{ee}(\mathbf{0})$, the temperature at which FM order
555Without loss of generality, we have assumed that magnetic order is
ferromagnetic. However, since we generally expect
$\chi_{ee}(\mathbf{q})\sim\rho$, in the weak-coupling regime the crossing
temperature is similar to the temperature at which magnetic order sets in
regardless of the type of order. sets in, before vanishing entirely, when
$T_{\rm RKKY}\gtrsim T_{K}$, the hybridisation field collapses and we have a
conventional RKKY phase with no easy-axis reorientation. Fig. 2 illustrates
the two regimes for both theory and the corresponding ratios experimentally
measured for a range of compounds. It is found that UCo1-xIrxGe spans the
range from strongly coupled ($T_{\rm N/C}/T_{\rm cross}\sim 0$) to weakly
coupled ($T_{\rm N/C}/T_{\rm cross}\sim 1$) as a function of the doping $x$,
where $T_{\rm N/C}$ is the Neel or Curie temperature depending on the type of
magnetic order. On the other hand, the Ce and Yb compounds tend to lie in the
weakly coupled regime, presumably because they are described by the fully
compensated Coqblin-Schrieffer model and the residual moments vanish when
$T_{K}\gg T_{\rm RKKY}$.

If we extrapolate the strong-coupling large-$N$ result to the physical case
$N=2$, we find that the crossing temperature is proportional to the strength
of the Kondo interaction, i.e. we should have $T_{\rm cross}\propto T^{*}_{\rm
coh}$, which indeed shows excellent agreement with experimental data collected
across a wide range of heavy-fermion compounds as shown in Fig. 1, where the
Kondo coherence temperature is varied with pressure or doping for three heavy
fermion compounds and is plotted against the easy-axis reorientation
temperature, $T_{\rm cross}$.

The above model would suggest that the axis with the lowest high-temperature
susceptibility (highest $D^{\eta}$) should always become the low-temperature
easy axis. This is rarely seen in orthorhombic systems (it has been seen in
YbRhSb Umeo  _et al._ (2012) at high pressures). In most compounds only the
two highest susceptibilities cross with the third one always remaining
quenched. We suggest that in those cases the anisotropy $D^{\eta}$ of the axis
with the lowest susceptibility is too large for the Kondo-hybridisation
mechanism to overcome. In fact, it usually has the lowest effective moment as
well as the lowest Curie-Weiss constant, which suggests that the mechanism is
not in play. $H_{\rm eff}$ is only valid for $D^{\eta}\ll T_{K}$, but if we
ramp up the anisotropy and look at the optimised variational wavefunction, the
hybridisation field that is responsible for the reduction of the effective
moment and the reinforcement of the Curie-Weiss constant eventually collapses
and we revert to free moment behaviour, where the higher the $D^{\eta}$, the
lower the susceptibility. This happens before the hybridisation field
collapses for the two directions with lower $D^{\eta}$. We note that, although
the collapse signals the return of the free-moment state, it is beyond the
validity of our perturbative model.

Maximum in the hard-axis susceptibility. –

Figure 3: The temperature of the b-axis maximum, $T^{b}_{\rm max}$, in UTe2
against the difference in the Curie-Weiss constants $\Theta^{a}_{\rm
CW}-\Theta^{b}_{\rm CW}$. The direct proportionlity fit, which does not
include the point above critical pressure with AFM order in the ground state,
shows that the exchange anisotropy is the energy scale that is responsible for
$T^{b}_{\rm max}$ at low pressures. The fit has $T^{b}_{\rm max}\approx
0.5(\Theta^{a}_{\rm CW}-\Theta^{b}_{\rm CW})$, close to the single-mode
classical prediction of $T^{b}_{\rm max}\approx 0.44(\Theta^{a}_{\rm
CW}-\Theta^{b}_{\rm CW})$. The inset shows the single-mode classical
simulation and experimental data at ambient pressure cited in the
acknowledgements.

A common observation in heavy-fermion compounds is the presence of a maximum
in the temperature dependence of the susceptibility along the low temperature
hard axis (high temperature easy axis)Li  _et al._ (2021). The maximum is
followed by a low-temperature region where $\left(\frac{\partial M}{\partial
T}\right)_{h}=-\left(\frac{\partial S}{\partial h}\right)_{T}<0$. Such a
maximum appears to be a necessary, but not sufficient, prerequisite for a
metamagnetic transition when a large field is applied along the low
temperature hard axis at even lower temperatures Miyake  _et al._ (2019);
Knebel  _et al._ (2024). Here we argue that the anisotropy in the exchange
that is responsible for the reorientation of the easy axis is also responsible
for the maximum in the hard-axis susceptibility. By the fluctuation-
dissipation theorem, the uniform susceptibility is a measure of the
$\mathbf{q}=\mathbf{0}$ fluctuations along a given axis
$\chi^{\eta}=\int_{0}^{\beta}\langle
S_{2}^{\eta}(\mathbf{0},\tau)S_{2}^{\eta}(\mathbf{0},0)\rangle_{H_{\rm
eff}}d\tau$. The cost of such fluctuations is lower along the easy axis than
the hard axis and can lead to a quenching of the hard-axis fluctuations and
susceptibility at a characteristic temperature that is of the order of the
energy difference between spins aligned along the easy axis and spins aligned
along the hard axis.

The quenching of the susceptibility can be captured by a single-mode classical
simulation and we will focus on the case of UTe2 at different pressures. We
assume that the spins are confined to the $ab$-plane because of strong
anisotropy along the $c$-direction (the $c$-axis susceptibility is
significantly lower in the relevant temperature range). The corresponding
classical Hamiltonian for spin-$S$ is given by

$\displaystyle H_{\rm cl}=-2S(S+1)\Delta\Theta_{\rm
CW}\sin^{2}\theta-\tilde{h}^{b}S\cos\theta,$

where $S=1/2$, only Goldstone $\mathbf{q}=\mathbf{0}$ fluctuations were
included and $\theta$ is the angle of the moment from the hard $b$-axis (field
axis) towards the easy $a$-axis. The temperature of the susceptibility maximum
is found to be $T^{b}_{\rm max}\approx 0.44(\Theta^{a}_{\rm
CW}-\Theta^{b}_{\rm CW})$, which is not far from the experimentally observed
trend $T^{b}_{\rm max}\approx 0.5(\Theta^{a}_{\rm CW}-\Theta^{b}_{\rm CW})$,
as shown in Fig. 3. (In the classical simulation, the effective field
$\tilde{h}^{b}$ was chosen to recover the high-temperature behaviour.) The
direct proportionality implies that exchange anisotropy is the energy scale
that is responsible for $T^{b}_{\rm max}$. We note that the single-mode
classical simulation underestimates the CW constant because AFM fluctuations
are not included. This is presumably also why $T^{b}_{\rm max}$ for $p>p_{c}$
(where AFM order sets in the ground state) departs from the direct
proportionality trend.

The relatively good agreement with the classical $S=1/2$ simulation adds to
the growing evidence that that far above the coherence temperature the local
moments in UTe2 are $S=1,L=0$ and, following Kondo-hybridisation, the residual
moments are $S=1/2,L=0$. Indeed, UTe2 has been argued to have a $5f^{2}$
valency Christovam  _et al._ (2024) and most recent measurements show the
effective moment to be approximately $2.8\,\mu_{\rm B}$ Ran  _et al._ (2019),
which is close to that of a free $S=1,L=0$ moment. The saturation
magnetisation per f.u. of $1\,\mu_{\rm B}$ at low temperatures points to a
spin-1/2 residual moment and a single screening channel. The situation in UTe2
should be compared with EuCu2(Ge1-xSix)2, where the local moments are believed
to be $S=7/2,L=0$ Iha  _et al._ (2018).

Spin-fluctuation mediated pairing interactions. –

Finally, we speculate on the possible pairing interactions of the heavy
fermions that could be mediated by the low-energy fluctuations of the residual
moments. In particular, we find a pairing interaction along the $\eta$-axis
between $f_{1}$ and $c$ fermions, which are now both part of the heavy-fermion
quasiparticle,

$\displaystyle H_{\rm pair}^{\eta}=-\frac{J_{K}}{N}\sum_{ij\eta\alpha\sigma}$
$\displaystyle
D^{\eta}\chi_{ij}^{\eta}\left(f^{\dagger}_{\alpha\sigma}(\mathbf{r}_{i})c^{\dagger}_{\alpha\sigma}(\mathbf{r}_{j})c_{\alpha\sigma}(\mathbf{r}_{j})f_{\alpha\sigma}(\mathbf{r}_{i})\right.$
$\displaystyle-\left.f^{\dagger}_{\alpha\sigma}(\mathbf{r}_{i})c^{\dagger}_{\alpha\bar{\sigma}}(\mathbf{r}_{j})c_{\alpha\bar{\sigma}}(\mathbf{r}_{j})f_{\alpha\sigma}(\mathbf{r}_{i}),\right)$
(14)

where the up/down fermionic spins are with respect to the $\eta$-axis and
$\chi_{ij}^{\eta}=\int_{0}^{\beta}\langle
S_{2\alpha}^{\eta}(\mathbf{r}_{i},\tau)S_{2\alpha}^{\eta}(\mathbf{r}_{j},0)\rangle_{H_{\rm
eff}}\;d\tau$ is the susceptibility of the residual $f_{2}$-moments. The above
interaction leads to the intriguing possibility of the pairing channel being
dependent on the sign of $D^{\eta}$. One might envisage a situation, where
fluctuations of the residual moments generate triplet pairing along the low-
temperature easy axis ($D^{\eta}>0$), whereas fluctuations of the residual
moments generate singlet pairing along the low-temperature hard axis
($D^{\eta}<0$). This could potentially be the scenario found in UTe2 for the
$a$ and $b$ axes respectively.

Conclusion. – In summary, we have highlighted universal behaviours in heavy-
fermion systems irrespective of the compound’s individual properties such as
the valency or crystal structure. We propose a large-$N$ theory which we
demonstrate captures the physics of these heavy-fermion systems. We identify
two regimes. When $T_{\rm N/C}/T_{\rm cross}\sim 1$ the reorientation of the
easy axis occurs at the onset of magnetic order and the system is in the weak-
coupling regime. The strong-coupling regime is characterised by $T_{\rm
cross}\propto T^{*}_{\rm coh}$. Many of the uranium and europium compounds
which we argue are in the strong-coupling regime exhibit a hierarchy of
decreasing energy scales: the coherence temperature $T^{*}_{\rm coh}$ and the
temperature of easy-axis reorientation $T_{\rm cross}$; the temperature of the
hard-axis maximum $T^{\rm hard}_{\rm max}$; the metamagnetic transition in
applied field along the hard-axis; and superconductivity.

We show that the easy-axis reorientation is caused by competing anisotropy in
the exchange and effective moment. Further, we demonstrate that the exchange
anisotropy can explain the maximum in the low temperature hard axis
susceptibility. A classical Hamiltonian that captures this maximum was derived
and agrees well with the experimental data.

Acknowledgments – We acknowledge the following people for providing
experimental data on ${\rm UCo}_{1-x}{\rm Ir}_{x}{\rm Ge}$ and ${\rm
URh}_{1-x}{\rm Ir}_{x}{\rm Ge}$: V. Sechovský, D. Hovančík, and J. Pospíšil
and F. Malte Grosche, A. Eaton, Z. Wu for providing experimental data on ${\rm
UTe}_{2}$. As well as those already mentioned, we would like to thank the
following people for stimulating discussions: P. Coleman, C. Batista, M.
Zhitomirsky, J. Annett, D. Adroja, F. Krüger, H. Hu, K. Wojcik, and G.
Lonzarich for many imaginative ideas.

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Compound | P (GPa) | ${x}$ |  $T^{*}_{\rm coh}$ (K) |  $T_{\rm cross}$ (K) |  $T_{\rm max}^{\rm hard}$ (K) |  $T_{N}$ (K) |  $T_{C}$ (K)
---|---|---|---|---|---|---|---
${\rm UTe_{2}}$ (Ortho.)Li  _et al._ (2021); Ajeesh  _et al._ (2024) | 0 |  | $108^{a\dagger}$ | 222 | 37 |  |
Crossing $b\rightarrow a$ | 0.6 |  | $81^{a\P}$ | 184 | 31 |  |
Metamagnetism observed along b at amb. p. | 0.9 |  | $65^{a\P}$ | 123 | 24 |  |
| 1.1 |  | $56^{a\P}$ | 108 | 22 |  |
| 1.4 |  | $42^{a\P}$ | 91 | 16 |  |
| 1.7 |  | $28^{a\P}$ | 45 |  | 3 |
${\rm UCo_{1-x}Ir_{x}Ge}$ (Ortho.)Hovančík  _et al._ (2022); R. Troć and Samsel-Czekała (2010) |  | $0.00$ | $74^{a}$ | $427^{\dagger\ddagger}$ |  |  | 2.5
Crossing $b\rightarrow c$ |  | 0.02 | $63^{a}$ | $359^{\dagger}$ |  |  |
Metamagnetism observed for $x>0.84$ along b |  | 0.07 | $40^{a}$ | $229$ |  |  |
|  | 0.24 | $24^{a}$ | $187$ | 35 |  |
|  | 0.40 | $24^{a}$ | 194 | 30 |  |
|  | 0.67 | $15^{a}$ | 45 | 25 |  |
|  | 0.84 | $25^{a}$ | 32 | 13 | 10.2 |
|  | 0.89 | $31^{c}$ | 27 | 18 | 12.2 |
|  | 0.92 | $29^{c}$ | 36 | 19 | 14.4 |
|  | 1.00 | $32^{c}$ | 21 | 29 | 16.5 |
${\rm URh_{1-x}Ir_{x}Ge}$ (Ortho.)Pospíšil  _et al._ (2017) |  | 0.00 | $188^{a}$ | – |  |  | 9.5
Crossing $b\rightarrow c$ |  | 0.14 | – | 69† |  |  | 9.1
Metamagnetism observed along b |  | 0.43 | – | – |  |  | 6.2
|  | 0.45 | – | – | $\sim 11$ | 3.9 |
|  | 0.58 | – | 34 | 12 | 7.0 |
|  | 1.00 | $32^{c}$ | 21 | 29 | 16.5 |
${\rm EuCu_{2}(Ge_{1-x}Si_{x})_{2}}$ (Tetra.)Iha  _et al._ (2018) |  | 0.6 | 0a§ | 0§ |  | 18.8 |
Crossing $a\rightarrow b$ |  | 0.7 | $55^{a}$ |  $\sim$ 60 |  | 15.5 |
|  | 0.8 | $142^{a}$ | 138 |  |  |
|  | 1.0 | $300^{a}$ | $>300\,?$ | $\sim 230$ |  |
${\rm UIr_{2}Si_{2}}$ (Tetra.)Szlawska  _et al._ (2023) |  |  | $97^{c}$ | 60 | 6 | 5.5 |
Crossing $a\rightarrow c$, |  |  |  |  |  |  |
${\rm UPt_{2}Si_{2}}$ (Tetra.)Bleckmann  _et al._ (2010) |  |  | $176^{a}$ | 76.5 |  | 32 |
Crossing $a\rightarrow c$ |  |  |  |  |  |  |
Metamagnetism observed along a Amitsuka  _et al._ (1992) |  |  |  |  |  |  |
${\rm UIrSi_{3}}$ (Tetra.)Valenta  _et al._ (2018); Honda  _et al._ (2019) |  |  | $224^{c}$ | 93 |  | 41.7 |
Crossing $a\rightarrow c$ |  |  |  |  |  |  |
${\rm EuNi_{2}P_{2}}$ (Tetra.)Hiranaka  _et al._ (2013) |  |  | $135^{a}$ | 76 | 46 |  |
Crossing $ab\rightarrow c$ |  |  |  |  |  |  |
Metamagnetism observed along ab |  |  |  |  |  |  |
YbNi4P2 (Tetra.)Hafner  _et al._ (2019); Krellner  _et al._ (2011); Steppke  _et al._ (2013) |  |  | $19^{c}$ | 0.16 |  |  | 0.15
Crossing $\parallel c\rightarrow\perp c$ |  |  |  |  |  |  |
CeRuPO (Tetra.)Hafner  _et al._ (2019); Krellner and Geibel (2008); Krellner  _et al._ (2007, 2008) |  |  | $41^{c}$ | $15^{*}$ |  |  | 15
Crossing $\perp c\rightarrow\parallel c$ |  |  |  |  |  |  |
Yb(Rh0.75Co0.25)2Si2 (Tetra.)Hafner  _et al._ (2019); Lausberg  _et al._ (2013); Klingner  _et al._ (2011); Gruner  _et al._ (2012) |  |  | $47^{c}$ | $1.3^{*}$ |  |  | 1.3
Crossing $\perp c\rightarrow\parallel c$ |  |  |  |  |  |  |
YbIr3Ge7 (Rhomb.)Hafner  _et al._ (2019); Rai  _et al._ (2019) |  |  | $7^{c}$ | 2.9 |  |  | 2.6
Crossing $\parallel c\rightarrow\perp c$ |  |  |  |  |  |  |
Table 1: A table of the coherence temperature $T^{*}_{\rm coh}$(K) defined as
the maximum in the resistivity; $T_{\rm cross}$(K), the temperature of the
easy axis reorientation; the temperature of the maximum in the low temperature
hard axis susceptibility; and $T_{C}$(K) or $T_{N}$(K) the Curie or Néel
temperature for compounds at various doping levels, $x$, or pressure, P (GPa).
The superscript on $T^{*}_{\rm coh}$ values indicate the axis for which the
resistivity measurement was taken.
$\ast$ the crossing occurs less than 1 K above the Curie temperature.
$\dagger$ indicates an extrapolation: mCW fits for $T_{\rm cross}$ or linear
extrapolation for $T^{*}_{\rm coh}$.
$\ddagger$ the crossing temperature was taken from Ref. R. Troć and Samsel-
Czekała (2010).
§ No susceptibility crossing or resistivity coherence maximum are observed
(this is true for all lower values of x in ${\rm
EuCu_{2}(Ge_{1-x}Si_{x})_{2}}$).
¶The coherence temperatures were determined from linear interpolations of
resistivity measurements at different pressure values.


# The disconnectedness of certain sets defined after uni-variate polynomials

Vladimir Petrov Kostov

###### Abstract.

We consider the set of monic real uni-variate polynomials of a given degree
$d$ with non-vanishing coefficients, with given signs of the coefficients and
with given quantities $pos$ of their positive and $neg$ of their negative
roots (all roots are distinct). For $d\geq 6$ and for signs of the
coefficients $(+,-,+,+,\ldots,+,+,-,+)$, we prove that the set of such
polynomials having two positive, $d-4$ negative and two complex conjugate
roots, is not connected. For $pos+neg\leq 3$ and for any $d$, we give the
exhaustive answer to the question for which signs of the coefficients there
exist polynomials with such values of $pos$ and $neg$.

Key words: real polynomial in one variable; hyperbolic polynomial; Descartes’
rule of signs; discriminant set

AMS classification: 26C10; 30C15

Author’s address: Université Côte d’Azur, CNRS, LJAD, France

email<EMAIL_ADDRESS>
##  1\. Introduction

We consider questions about the general family of monic uni-variate real
degree $d$ polynomials: $Q_{d}:=x^{d}+\sum_{j=0}^{d-1}a_{j}x^{j}$. In the
space $\mathbb{R}^{d}$ of the coefficients $a_{j}$ one defines the
discriminant set $\Delta_{d}$ as the set of their values for which the
polynomial $Q_{d}$ has a multiple real root. More precisely, if
$\Delta^{1}_{d}$ is the set of values of the coefficients for which $Q_{d}$
has a multiple root (real or complex), then this is the set of the zeros of
the determinant of the Sylvester matrix of the polynomials $Q_{d}$ and
$Q_{d}^{\prime}$. One has to set
$\Delta_{d}:=\Delta_{d}^{1}\setminus\Delta_{d}^{2}$, where $\Delta_{d}^{2}$ is
the set of values of the coefficients $a_{j}$ for which there is a multiple
complex conjugate pair of roots of $Q_{d}$ and no multiple real root. It is
true that dim$(\Delta_{d})=$dim$(\Delta_{d}^{1})=d-1$ and
dim$(\Delta^{2}_{d})=d-2$.

The set

$R_{1,d}:=\mathbb{R}^{d}\setminus\Delta_{d}$

consists of $[d/2]+1$ open components of dimension $d$ ($[.]$ stands for the
integer part of). The polynomials $Q_{d}$ from a given component have one and
the same number $\mu$ of real roots (which are all distinct); the number $\nu$
of complex conjugate pairs can range from $0$ to $[d/2]$, because
$\mu+2\nu=d$. Given two polynomials with one and the same number $\nu$, one
can continuously deform the roots of the first polynomial into the roots of
the second one by keeping the real roots distinct throughout the deformation.
This proves that to any possible number $\nu$ corresponds exactly one
component of the set $R_{1,d}$.

In the same way one can consider the components of the set

$R_{2,d}:=\mathbb{R}^{d}\setminus(\Delta_{d}\cup\\{a_{0}=0\\})~{}.$

The polynomials from one and the same open component (also of dimension $d$)
have one and the same numbers $pos$ of positive and $neg$ of negative roots
(and no vanishing roots). When deforming the roots of one polynomial into the
roots of another one, one has to keep the same numbers $pos$ and $neg$
throughout the deformation. To each pair ($pos$, $neg$) corresponds exactly
one component of the set $R_{2,d}$. As $pos+neg=\mu$, $0\leq pos$,
$neg\leq\mu$ and $\mu+2\nu=d$, there are

$(d+1)+(d-1)+(d-3)+\cdots=([d/2]+1)([(d+1)/2]+1)$

components of the set $R_{2,d}$.

A more complicated task is to study the components of the set

$R_{3,d}:=\mathbb{R}^{d}\setminus(\Delta_{d}\cup\\{a_{0}=0\\}\cup\\{a_{1}=0\\}\cup\cdots\cup\\{a_{d-1}=0\\})$

of monic uni-variate polynomials with no multiple real roots and no zero
coefficients.

######  Definition 1.

A sign pattern of length $d+1$ is a sequence of $d+1$ symbols $+$ and/or $-$
beginning with a $+$. We say that a polynomial $Q_{d}$ with no vanishing
coefficients defines the sign pattern
$\sigma_{0}:=(+,\beta_{d-1},\beta_{d-2},\ldots,\beta_{0})$, $\beta_{j}=+$ or
$-$, (notation: $\sigma(Q_{d})=\sigma_{0}$), if sign$(a_{j})=\beta_{j}$,
$j=0$, $\ldots$, $d-1$.

One can ask the question to which couple (sign pattern, pair $(pos,neg)$) (we
call them couples for short) corresponds at least one component of the set
$R_{3,d}$. The polynomials from a given component of $R_{3,d}$ have one and
the same couple. All components are of dimension $d$.

When considering the set $R_{3,d}$, it is self-understood that the couples
have to be defined in accordance with Descartes’ rule of signs. This rule
states that a real uni-variate polynomial $Q_{d}$ has not more positive roots
counted with multiplicity than the number $c$ of sign changes in the sequence
of its coefficients; the difference $c-pos$ is even, see [4], [9], [10], [11],
[15], [16], [19], [20], [28] or [30]. Hence the sign of the constant term is
$(-1)^{pos}$. When the polynomial has no zero coefficients, Descartes’ rule of
signs applied to $Q_{d}(-x)$ implies that $Q_{d}$ has not more negative roots
counted with multiplicity than the number $p$ of sign preservations in that
sequence (hence $c+p=d+1$), and the difference $p-neg$ is also even.

######  Definition 2.

A pair $(pos,neg)$ satisfying these conditions w.r.t. a given sign pattern
$\sigma_{0}$ is called compatible with $\sigma_{0}$ (and vice versa), and the
couple $(\sigma_{0},(pos,neg))$ is also called compatible. For a monic
polynomial $Q_{d}$ with no vanishing coefficients, with $pos$ positive simple
and $neg$ negative simple roots and no other real roots, we say that $Q_{d}$
realizes the couple $(\sigma(Q_{d}),(pos,neg))$.

Yet this compatibility is just a necessary condition which turns out not to be
sufficient. That is, there exist cases when to certain compatible couples
correspond no components of $R_{3,d}$. So we formulate the first problem which
we consider in the present paper:

######  Problem 1.

For a given degree $d$, for which compatible couples do there exist monic
polynomials realizing these couples? In other words, to which of the
compatible couples there corresponds at least one component of the set
$R_{3,d}$?

Some results in relationship with Problem 1 are formulated in the next
section. The problem seems to have been stated for the first time in [2]. The
first example when to a compatible couple corresponds no component of the set
$R_{3,d}$ (this is an example with $d=4$), and the exhaustive answer to the
problem for $d=4$, are to be found in [18]. For $d=5$ and $d=6$, the result is
given in [1]. For $d=7$ and partially for $d=8$ (resp. completely for $d=8$),
the answer is formulated and proved in [12] and [13] (resp. in [22]).
Different aspects concerning Descartes’ rule of signs are treated in papers
[23], [5], [6], [7], [8] and [14].

Of particular importance is the class of hyperbolic polynomials, i. e. real
polynomials whose roots are all real. The hyperbolicity domain $\Pi_{d}$ is
the set of values of the coefficients $a_{j}$ for which the polynomial $Q_{d}$
is hyperbolic. For properties of hyperbolic polynomials and the domain
$\Pi_{d}$ see [3], [17], [21], [29] and [24].

In what follows we are also interested in another problem:

######  Problem 2.

For a given degree $d$, to which compatible couples correspond two or more
components of the set $R_{3,d}$?

To formulate our first result connected with Problem 2 we introduce the
following notation:

######  Notation 1.

For $d\geq 4$, we consider $\mathbb{R}^{d}$ as the set
$\\{(a_{d-1},a_{d-2},\ldots,a_{0})|a_{j}\in\mathbb{R}\\}$ of $d$-tuples of
coefficients (excluding the leading one) of polynomials $Q_{d}$. We denote by
$\sigma_{\bullet}$ the sign pattern $(+,-,+,+,\ldots,+,+,-,+)$ and by
$\Pi^{*}_{d}(\sigma_{\bullet})$ (resp. by $A(\sigma_{\bullet},(2,d-4))$) the
subset of $\mathbb{R}^{d}$ of polynomials with signs of the coefficients (all
non-zero) as defined by $\sigma_{\bullet}$ and having four positive and $d-4$
negative distinct real roots (resp. two positive and $d-4$ negative distinct
roots and one complex conjugate pair). Hence the polynomials of the set
$\Pi^{*}_{d}(\sigma_{\bullet})$ are hyperbolic while the ones of the set
$A(\sigma_{\bullet},(2,d-4))$ are not.

The following theorem is proved in Section 3.

######  Theorem 1.

(1) For $d\geq 6$, the set $A(\sigma_{\bullet},(2,d-4))$ is non-empty and
consists of more than one component of the set $R_{3,d}$. Hence the set
$A(\sigma_{\bullet},(2,d-4))$ is not connected.

(2) For $d=4$ and $5$, the respective sets $A(\sigma_{\bullet},(2,0))$ and
$A(\sigma_{\bullet},(2,1))$ are connected.

######  Remarks 1.

(1) One can mention cases in which the components of the set $R_{3,d}$ are
contractible and to each compatible couple corresponds exactly one component
of the set $R_{3,d}$ (see [26]). Namely, such are the cases of hyperbolic
polynomials and of polynomials having exactly one or no real roots at all.

(2) In the case of polynomials having exactly two real distinct roots (hence
$pos+neg=2$) to each compatible couple corresponds either one or no component
of $R_{3,d}$, and all components are contractible. See more details in the
next section or in [26]. Whether in the case of exactly three real roots to
each compatible couple corresponds at most one component of the set $R_{3,d}$
is an open question.

(3) For $d=4$ and $d=5$, pictures of the set $\Delta_{d}^{1}$ (from which one
can deduce the form of the set $A(\sigma_{\bullet},(2,d-4))$) can be found in
[27] and [8] respectively.

##  2\. Comments and further results

Given a sign pattern $\hat{\sigma}$ with $c$ sign changes and $p$ sign
preservations (hence $c+p=d$), Descartes’ rule of signs implies that any
hyperbolic polynomial with sign pattern $\hat{\sigma}$ has exactly $c$
positive and exactly $p$ negative roots counted with multiplicity. We define
the canonical order of moduli corresponding to $\hat{\sigma}$. The sign
pattern $\hat{\sigma}$ is read from the right and to each sign change (resp.
sign preservation) one puts in correspondence the letter $P$ (resp. the letter
$N$).

For example, for $\hat{\sigma}=\sigma_{\dagger}:=(+,-,-,-,+,+)$ (resp. for
$\hat{\sigma}=\sigma_{\bullet}$) this gives the string $NPNNP$ (resp.
$PPNN\cdots NNPP$, $d-4$ times $N$). After this one inserts the symbol $<$
between any two consecutive letters which in the cases of $\sigma_{\dagger}$
and $\sigma_{\bullet}$ gives

$N<P<N<N<P~{}~{}~{}\,{\rm and}~{}~{}~{}\,P<P<N<N<\cdots<N<N<P<P$

respectively. If one denotes by $\alpha_{j}$ and $\beta_{j}$ the moduli of the
positive and negative roots, then one replaces the letters $P$ and $N$ by
these moduli which in the case of $\sigma_{\dagger}$ defines the canonical
order

$\beta_{1}<\alpha_{1}<\beta_{2}<\beta_{3}<\alpha_{2}$

whereas the canonical order corresponding to $\sigma_{\bullet}$ is given by
(3.1).

It is true that for any sign pattern $\sigma_{0}$ of length $d+1$, there
exists a degree $d$ monic hyperbolic polynomial $T$ with
$\sigma(T)=\sigma_{0}$ whose roots define the respective canonical order of
moduli, see Proposition 1 in [25].

Our next step is to consider the cases when the polynomial $Q_{d}$ has not
more than three real roots, i. e. $pos+neg\leq 3$ (and hence in the case of
equality the possible values of the pair $(pos,neg)$ are $(3,0)$, $(2,1)$,
$(1,2)$ and $(0,3)$). For the cases $pos=neg=0$ and $pos+neg=1$, see part (1)
of Remarks 1. For $pos+neg=2$ (hence $d$ is even), we remind some of the
results of [26].

######  Definition 3.

For $pos+neg=2$, we define Case 1) (resp. Case 2)) by the conditions the
constant term to be positive, all coefficients of monomials of odd degree to
be positive (resp. negative), the pair $(pos,neg)$ to equal $(2,0)$ (resp.
$(0,2)$) and the coefficient of at least one monomial of even degree to be
negative.

######  Theorem 2.

(see [26]). For $d$ even and $pos+neg=2$,

(1) A given compatible couple is realizable if and only if it does not
correspond to Case 1) or 2).

(2) If the constant term is positive (hence $(pos,neg)=(2,0)$ or $(0,2)$) and
one is not in Case 1) or 2), a given compatible couple is realizable by
polynomials having any ratio different from $1$ between the moduli of the two
real roots.

(3) If the constant term is negative (hence $(pos,neg)=(1,1)$) and there are
two monomials of odd degree with coefficients of opposite signs, then such a
compatible couple is realizable by polynomials with any ratio of the moduli
$\alpha$ and $\beta$ of its positive and negative root respectively.

(4) If the constant term is negative and all coefficients of monomials of odd
degree are positive (resp. negative), then such a compatible couple is
realizable by polynomials with any ratio $\alpha/\beta<1$ (resp.
$\alpha/\beta>1$) and not realizable by polynomials with $\alpha/\beta\geq 1$
(resp. $\alpha/\beta\leq 1$).

To formulate the new results about the situation with $pos+neg=3$ we introduce
the following notion:

######  Definition 4.

For a given degree $d$, the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$-action on the
set of compatible couples is defined by two commuting involutions. The first
of them maps a polynomial $Q_{d}$ into $(-1)^{d}Q_{d}(-x)$ (this changes the
pair $(pos,neg)$ into $(neg,pos)$, it changes the signs of the coefficients of
$x^{d-1}$, $x^{d-3}$, $\ldots$ and preserves the signs of the other
coefficients). The second involution maps $Q_{d}$ into
$x^{d}Q_{d}(1/x)/Q_{d}(0)$ (the pair $(pos,neg)$ is preserved and the sign
pattern, eventually multiplied by $-1$, is read from the right; the roots of
$x^{d}Q_{d}(1/x)/Q_{d}(0)$ are the reciprocals of the roots of $Q_{d}$). An
orbit of the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$-action consists of $2$ or
$4$ compatible couples which are simultaneously realizable or not. This allows
to formulate the results only for one of the $2$ or $4$ couples of a given
orbit.

######  Theorem 3.

Suppose that the pair $(2,1)$ is compatible with the sign pattern
$\sigma_{\triangle}$ (hence the constant term is positive). Then

(1) The couple $\mathcal{C}:=(\sigma_{\triangle},(2,1))$ is realizable.

Denote by $-\beta<0$, $\alpha_{1}>0$ and $\alpha_{2}>0$ the three real roots
of a polynomial realizing the couple $\mathcal{C}$.

(2) If there are monomials $x^{2m}$ and $x^{2n-1}$ with negative coefficients
(one can have $2m<2n-1$ or $2n-1<2m$), then for any of the five possibilities

$\beta<\alpha_{1}<\alpha_{2}~{},~{}~{}~{}\,\,\beta=\alpha_{1}<\alpha_{2}~{},~{}~{}~{}\,\,\alpha_{1}<\beta<\alpha_{2}~{},~{}~{}~{}\,\,\alpha_{1}<\alpha_{2}=\beta~{}~{}~{}\,\,{\rm
and}~{}~{}~{}\,\,\alpha_{1}<\alpha_{2}<\beta~{},$

there exist polynomials realizing the couple $\mathcal{C}$.

(3) If all odd monomials have positive coefficients, then only the possibility
$\beta<\alpha_{1}<\alpha_{2}$ is realizable.

(4) If all even monomials have positive coefficients, then only the
possibility $\alpha_{1}<\alpha_{2}<\beta$ is realizable.

The theorem is proved in Section 4. The compatibility of the sign pattern with
the pair $(2,1)$ implies that in part (3) (resp. in part (4)) of the theorem
there is at least one even (resp. odd) monomial whose coefficient is negative.

######  Notation 2.

For $d$ odd, we denote by $D(a,b,c)$ the sign pattern consisting of $2a$
pluses followed by $b$ pairs “$-,+$” followed by $2c$ minuses, where $1\leq
a$, $1\leq b$, $1\leq c$ and $2a+2b+2c=d+1$.

######  Theorem 4.

Suppose that the pair $(3,0)$ is compatible with the sign pattern
$\sigma_{\diamond}$ which is not of the form $D(a,b,c)$. Then the couple
$(\sigma_{\diamond},(3,0))$ is realizable.

The theorem is proved in Section 5.

######  Theorem 5.

For $j=1$, $2$, $\ldots$, $b$, the couple $(D(a,b,c),(2j+1,0))$ is not
realizable.

The theorem is proved in Section 6. Its proof resembles the proof of part (i)
of Theorem 4 in [27] which treats a particular case of Theorem 5. However the
proof of Lemma 1 (used in the proof of Theorem 5) is more complicated than the
proof of its analog which is Lemma 6 of [27]. This renders indispensable
giving the whole proof of Theorem 5.

######  Remark 1.

For the sign pattern $D(a,b,c)$, compatible are the following pairs
$(pos,neg)$:

1) the ones mentioned in Theorem 5;

2) the pair $(1,0)$;

3) the pairs $(2j+1,2r)$, $r=1$, $2$, $\ldots$, $a+c-1$, $j=0$, $1$, $\ldots$,
$b$.

Realizability of the couples $(D(a,b,c),(pos,neg))$ with $(pos,neg)$ as in 2)
and 3) can be proved by analogy with the proof of parts (ii) and (iii) of
Theorem 4 in [27].

##  3\. Proof of Theorem 1

###### Part (1).

A) For $d\geq 6$, the set $\Pi^{*}_{d}(\sigma_{\bullet})$ is non-empty, see
Proposition 1 in [25]. Fix a polynomial
$Q^{*}\in\Pi^{*}_{d}(\sigma_{\bullet})$. By Proposition 1 of [25], one can
choose $Q^{*}$ such that the moduli of its positive and negative roots
(denoted by $\alpha_{1}<\alpha_{2}<\alpha_{3}<\alpha_{4}$ and
$\beta_{1}<\beta_{2}<\cdots<\beta_{d-5}<\beta_{d-4}$ respectively) satisfy the
string of inequalities

(3.1)
$\alpha_{1}<\alpha_{2}<\beta_{1}<\beta_{2}<\cdots<\beta_{d-5}<\beta_{d-4}<\alpha_{3}<\alpha_{4}~{}.$

So the negative roots of $Q^{*}$ are
$-\beta_{d-4}<-\beta_{d-5}<\cdots<-\beta_{1}<0$. Starting with $Q^{*}$, we
construct two polynomials $Q^{1}$ and $Q^{2}$ of the set
$A(\sigma_{\bullet},(2,d-4))$ (so this set is non-empty) about which we show
that they belong to different components of $R_{3,d}$. This implies the
theorem.

B) We consider the one-parameter family of polynomials

$\tilde{Q}_{t}:=Q^{*}+tx^{2}(x+\beta_{1})(x+\beta_{2})\cdots(x+\beta_{d-4})~{},~{}~{}~{}t\geq
0~{}.$

For any $t\geq 0$, one has $\sigma(\tilde{Q}_{t})=\sigma_{\bullet}$. As $t$
increases, the roots $-\beta_{1}$, $-\beta_{2}$, $\ldots$, $-\beta_{d-4}$ of
$\tilde{Q}_{t}$ do not move. The roots $\alpha_{1}$ and $\alpha_{3}$ move to
the right while $\alpha_{2}$ and $\alpha_{4}$ move to the left. For some
$t_{0}>0$, either $\alpha_{1}$ coalesces with $\alpha_{2}$ or $\alpha_{3}$
coalesces with $\alpha_{4}$ or both these things take place. Indeed, the
values of $\tilde{Q}_{t}$ for each fixed $x\geq\alpha_{1}$ increase at least
as fast as $t\alpha_{1}^{2}\prod_{i=1}^{d-4}(\alpha_{1}+\beta_{i})$.

If for $t=t_{0}$, $\alpha_{1}$ and $\alpha_{2}$ coalesce and $\alpha_{3}$ and
$\alpha_{4}$ remain positive and distinct, then one can fix $t_{1}>t_{0}$
sufficiently close to $t_{0}$ for which the roots $\alpha_{1}$ and
$\alpha_{2}$ have given birth to a complex conjugate pair while $\alpha_{3}$
and $\alpha_{4}$ are still positive and distinct. We set
$Q^{1}:=\tilde{Q}_{t_{1}}$. Hence the polynomial $Q^{1}$ has $d-2$ real roots

(3.2)
$\begin{array}[]{ll}-\beta_{d-4}<-\beta_{d-5}<\cdots<-\beta_{1}<0<\alpha_{3}<\alpha_{4}&{\rm
such~{}that}\\\ \\\
0<\beta_{1}<\beta_{2}<\cdots<\beta_{d-4}<\alpha_{3}<\alpha_{4}&\end{array}$

and a complex conjugate pair. After this we set $Q^{2}_{*}:=x^{d}Q^{1}(1/x)$.
The sequence of coefficients of $Q^{1}$, when read from the right, is the
string of coefficients of $Q^{2}_{*}$. After this we set
$Q^{2}:=Q^{2}_{*}/Q^{1}(0)$, so $Q^{2}$ is monic. The sign pattern
$\sigma_{\bullet}$ is center-symmetric, therefore
$\sigma(Q^{2})=\sigma_{\bullet}=\sigma(Q^{1})$. The roots of the polynomial
$Q^{2}$ are the reciprocals of the roots of $Q^{1}$. The real roots of $Q^{2}$
satisfy the conditions

(3.3)
$\begin{array}[]{ll}-\beta_{d-4}<-\beta_{d-5}<\cdots<-\beta_{1}<0<\alpha_{3}<\alpha_{4}&{\rm
and}\\\ \\\
0<\alpha_{1}<\alpha_{2}<\beta_{1}<\beta_{2}<\cdots<\beta_{d-4};\end{array}$

the polynomial $Q^{2}$ has also a complex conjugate pair.

If for $t=t_{0}$, $\alpha_{3}$ and $\alpha_{4}$ coalesce while $\alpha_{1}$
and $\alpha_{2}$ remain positive and distinct, then for some $t_{1}>t_{0}$
sufficiently close to $t_{0}$ we obtain the polynomial $Q^{2}$ with exactly
two positive and $d-4$ negative roots which satisfy conditions (3.3). After
this we set $Q^{1}_{*}=x^{d}Q^{2}(1/x)$ and $Q^{1}:=Q^{1}_{*}/Q^{2}(0)$. The
real roots of $Q^{1}$ satisfy conditions (3.2).

Finally, if for $t=t_{0}$, one has $\alpha_{1}=\alpha_{2}=a>0$ and
$\alpha_{3}=\alpha_{4}=b>a$, then one constructs the polynomials

$Q^{\pm}:=\tilde{Q}_{t_{0}}\pm\varepsilon(x-(a+b)/2)~{},~{}~{}~{}\varepsilon>0~{}.$

For $\varepsilon$ small enough,

1) the coefficients of $Q^{\pm}$ are non-zero and
$\sigma(Q^{\pm})=\sigma_{\bullet}$;

2) each of the polynomials $Q^{\pm}$ has $d-4$ distinct negative roots close
to $-\beta_{i}$;

3) $Q^{+}$ has two distinct positive roots close to $a$ and a complex
conjugate pair close to $b$;

4) and vice versa for $Q^{-}$.

We set $Q^{1}:=Q^{-}$ and $Q^{2}:=Q^{+}$.

C) Suppose that the two polynomials $Q^{1}$ and $Q^{2}$ belong to one and the
same component of the set $R_{3,6}$. Then it is possible to connect them by a
continuous path (homotopy) within this component: $Q^{s}$, $s\in[1,2]$. Along
the path the two positive, the $d-4$ negative and the two complex conjugate
roots of $Q^{s}$ depend continuously on $s$ while remaining distinct
throughout the homotopy. We denote the negative roots by $-\tilde{\beta}_{j}$,
$j=1$, $\ldots$, $d-4$, and the two positive roots by $\tilde{\gamma}_{j}$,
$j=1$, $2$, where

$\begin{array}[]{lllll}{\rm for{}{}{}}&s=1~{},&{\rm
one~{}has}&\tilde{\beta}_{j}=\beta_{j}~{},&\tilde{\gamma}_{j}=\alpha_{2+j}~{};\\\
\\\ {\rm for{}{}{}}&s=2~{},&{\rm
one~{}has}&\tilde{\beta}_{j}=\beta_{j}~{},&\tilde{\gamma}_{j}=\alpha_{j}~{}.\end{array}$

Hence there exists $s=s_{0}\in(1,2)$ such that for $s=s_{0}$,
$\tilde{\beta}_{d-4}=\tilde{\gamma}_{2}$. This means that the polynomial
$Q^{s_{0}}$ has exactly $d-2$ real roots such that

$-\tilde{\beta}_{d-4}<\cdots<-\tilde{\beta}_{1}<0<\tilde{\gamma}_{1}<\tilde{\gamma}_{2}~{}~{}~{},~{}~{}~{}\,\,\tilde{\beta}_{d-4}=\tilde{\gamma}_{2}~{}.$

Using a linear change $x\mapsto hx$, $h>0$, we achieve the condition
$\tilde{\beta}_{d-4}=\tilde{\gamma}_{2}=1$.

D) Suppose that $d$ is even. The fact that $\pm 1$ are roots of $Q_{d}$
implies the two conditions:

$a_{d}+a_{d-2}+a_{d-4}+\cdots+a_{2}+a_{0}=0~{}~{}~{}\,\,{\rm
and}~{}~{}~{}\,\,a_{d-1}+a_{d-3}+\cdots+a_{3}+a_{1}=0~{}.$

The first of them is possible only if all even coefficients are $0$, because
in the corresponding positions the sign pattern $\sigma_{\bullet}$ contains
$(+)$-signs. However $a_{d}=1$. This contradiction means that the homotopy
$Q^{s}$ does not exist, so $Q^{1}$ and $Q^{2}$ belong to different components
of the set $R_{3,d}$ and the set $A(\sigma_{\bullet},(2,d-4))$ is not
connected. One can observe that this resoning is not valid for $d=2$ or $d=4$,
because in these cases there are no negative roots at all.

E) Suppose that $d\geq 7$ is odd. Set $\delta:=\tilde{\beta}_{d-5}>0$ and
$Q_{d}^{s_{0}}=(x+\delta)U(x)$, where $U=x^{d-1}+\sum_{j=0}^{d-2}u_{j}x^{j}$.
The polynomial $U$ has an even number of positive roots, so $u_{0}>0$. The
conditions

$0>a_{d-1}=\delta+u_{d-2}~{}~{}~{}\,\,{\rm and}~{}~{}~{}\,\,\delta>0$

imply $u_{d-2}<0$ whereas from

$0<a_{d-2}=\delta u_{d-2}+u_{d-3}~{},~{}~{}~{}\delta>0~{}~{}~{}\,\,{\rm
and}~{}~{}~{}\,\,u_{d-2}<0$

one deduces that $u_{d-3}>0$. In the same way one has

$\begin{array}[]{ll}0>a_{1}=\delta
u_{1}+u_{0}~{},~{}~{}~{}\delta>0~{},~{}~{}~{}u_{0}>0~{},~{}~{}~{}\,\,{\rm
so}~{}~{}~{}\,\,u_{1}<0&{\rm and}\\\ \\\ 0<a_{2}=\delta
u_{2}+u_{1}~{},~{}~{}~{}\delta>0~{},~{}~{}~{}u_{1}<0~{},~{}~{}~{}\,\,{\rm
so}~{}~{}~{}\,\,u_{2}>0~{}.&\end{array}$

The first three and the last three of the coefficients of the polynomial
$U(-x)$ are positive. By Descartes’ rule of signs it has not more than $d-5$
positive roots, and it has exactly $d-5$ positive roots only if it has $d-5$
sign changes. On the other hand one knows that $U(-x)$ has exactly $d-5$
positive roots $-\tilde{\beta}_{j}$, $j=1$, $2$, $\ldots$, $d-6$, $d-4$. Hence
$U(x)$ has $d-5$ sign preservations, therefore $u_{k}>0$ for $2\leq k\leq
d-3$.

Thus $\sigma(U)=\sigma_{\bullet}$ (but here the sign pattern
$\sigma_{\bullet}$ is meant to be of length $d$, not $d+1$). Suppose that the
homotopy $Q^{s}$ exists. Along this homotopy the root $-\tilde{\beta}_{d-5}$
is a continuous negative-valued function. As division of $Q^{s}$ by $x+\delta$
gives the polynomials $U$, there exists a homotopy between the polynomial $U$
corresponding to $Q^{1}$ and the one corresponding to $Q^{2}$. We denote them
by $U^{1}$ and $U^{2}$. They are of even degree $d-1\geq 6$, each of them has
exactly two positive roots $\tilde{\gamma}_{1}<\tilde{\gamma}_{2}$, exactly
$d-5$ negative roots and one complex conjugate pair. For the moduli of the
real roots one has

$\tilde{\beta}_{j}<\tilde{\gamma}_{1}~{}~{}~{}\,\,{\rm
for}~{}~{}~{}\,\,U^{1}~{}~{}~{}\,\,{\rm
and}~{}~{}~{}\,\,\tilde{\gamma}_{2}<\tilde{\beta}_{j}~{}~{}~{}\,\,{\rm
for}~{}~{}~{}\,\,U^{2}~{},~{}~{}~{}j=1,~{}2,\ldots~{},~{}d-6,~{}d-4$

(see (3.2) and (3.3)). This, however, is impossible, see D). ∎

###### Part (2).

F) For $d=4$, for each polynomial $Q\in A(\sigma_{\bullet},(2,0))$, there
exists a unique quantity $g>0$ such that for $g^{\prime}\in[0,g)$, one has
$Q+g^{\prime}\in A(\sigma_{\bullet},(2,0))$ and for $g^{\prime}=g$, the
polynomial $Q+g^{\prime}$ has a multiple positive root.

On the other hand, for each polynomial $Q\in A(\sigma_{\bullet},(2,0))$, there
exists a unique quantity $h>0$ such that for $h^{\prime}\in[0,h)$, one has
$Q-h^{\prime}\in A(\sigma_{\bullet},(2,0))$ and for $h^{\prime}=h$, $Q$ has
either a zero root or a multiple positive root. The quantities $g$ and $h$ are
continuous functions of the coefficients of $Q$.

Denote by $A^{*}(\sigma_{\bullet})$ the set of monic polynomials whose
coefficients have signs as defined by the sign pattern $\sigma_{\bullet}$ and
which have a multiple positive root and a complex conjugate pair. Hence the
set $A(\sigma_{\bullet},(2,0))$ is homeomorphic to the direct product of the
set $A^{*}(\sigma_{\bullet})$ and an open interval. Therefore if
$A^{*}(\sigma_{\bullet})$ is connected, then such is
$A(\sigma_{\bullet},(2,0))$ as well.

Denote by $A^{*}_{0}(\sigma_{\bullet})$ the subset of
$A^{*}(\sigma_{\bullet})$ for which the multiple root of $Q$ is at $1$. Each
polynomial $Q\in A^{*}(\sigma_{\bullet})$ can be transformed into a polynomial
of $A^{*}_{0}(\sigma_{\bullet})$ by a linear change of the variable $x$
followed by a multiplication with a non-zero constant. Hence
$A^{*}(\sigma_{\bullet})$ is homeomorphic to
$A^{*}_{0}(\sigma_{\bullet})\times(0,\infty)$.

Any polynomial $Q\in A^{*}_{0}(\sigma_{\bullet})$ is of the form

$(x-1)^{2}(x^{2}+Ax+B)=x^{4}+(A-2)x^{3}+(B-2A+1)x^{2}+(A-2B)x+B~{},$

where $A^{2}-4B\leq 0$. The set $A^{*}_{0}(\sigma_{\bullet})$ is defined by
the conditions

$A<2~{},~{}~{}~{}\,\,B-2A+1>0~{},~{}~{}~{}\,\,A-2B<0~{}~{}~{}\,\,{\rm
and}~{}~{}~{}\,\,B\geq A^{2}/4~{}.$

This is the set of points in the plane $(A,B)$ which are to the left of the
vertical line $A=2$ and above or on the graph of the function (of the argument
$A\in(-\infty,2)$) $\max(2A-1,A/2,A^{2}/4)$; strictly above for $A\in[0,2)$
and above or on the graph for $A<0$. This is a contractible set.

G) For $d=5$, we denote by $A^{\dagger}(\sigma_{\bullet})$ the set of monic
polynomials the signs of whose coefficients are defined by the sign pattern
$\sigma_{\bullet}$ and which have a simple negative root, a double positive
root and a complex conjugate pair. Denote by
$A^{\dagger}_{0}(\sigma_{\bullet})$ its subset for which the double root is at
$1$. By complete analogy with part F) of the proof we show that connectedness
of $A^{\dagger}_{0}(\sigma_{\bullet})$ implies the one of
$A(\sigma_{\bullet},(2,1))$.

Any polynomial $Q\in A^{\dagger}_{0}(\sigma_{\bullet})$ is of the form

$(x-1)^{2}(x+A)(x^{2}+Bx+C)=x^{5}+\sum_{j=0}^{4}f_{j}x^{j}~{},~{}~{}~{}\,\,{\rm
where}$

$\begin{array}[]{ll}f_{4}=A+B-2~{},&f_{3}=AB-2A-2B+C+1~{},\\\ \\\
f_{2}=-2AB+AC+A+B-2C~{},&f_{1}=AB-2AC+C\\\ \\\ {\rm
and}&f_{0}=AC~{},\end{array}$

with $A>0$ and $B^{2}-4C<0$. For any $\rho>0$ and $r>0$, the polynomial
$Q_{\rho,r}:=Q+\rho(x-1)^{2}+rx^{3}(x-1)^{2}$ defines the sign pattern
$\sigma_{\bullet}$ and belongs to the set $A^{\dagger}_{0}(\sigma_{\bullet})$.
Indeed, it is non-negative for $x\geq 0$, with equality only for $x=1$; its
second derivative at $x=1$ is positive, so $x=1$ is a double root; the sign
pattern $\sigma_{\bullet}$ and Descartes’ rule of sign imply that $Q_{\rho,r}$
has not more than one negative root, so it has exactly one such root. Hence
one can choose $\rho$ and $r$ such that $f_{2}=f_{3}$. The set
$A^{\dagger}_{0}(\sigma_{\bullet})$ is connected if and only if its subset
defined by the condition $f_{2}=f_{3}$ is connected.

H) The condition $f_{2}=f_{3}$ allows to express $A$ as a function of $B$ and
$C$:

$A=T_{0}/D~{},~{}~{}~{}\,\,{\rm
where}~{}~{}~{}\,\,T_{0}=3B-3C-1~{}~{}~{}\,\,{\rm
and}~{}~{}~{}\,\,D=3B-C-3~{}.$

For the coefficients $f_{i}$ with $A=T_{0}/D$ one finds

$\begin{array}[]{ll}f_{4}=T_{4}/D~{},&T_{4}=3B^{2}-BC-6B-C+5~{},\\\ \\\
f_{3}=f_{2}=T_{3}/D~{},&T_{3}=-3B^{2}+2BC-C^{2}+2B+2C-1~{},\\\ \\\
f_{1}=T_{1}/D~{},&T_{1}=3B^{2}-6BC+5C^{2}-B-C~{}.\end{array}$

In Fig. 1 and 2 we represent the following sets:

– $\mathcal{L}_{0}:T_{0}=0$ (in solid line) and $\mathcal{L}:D=0$ (in dashed
line) are straight lines;

– $\mathcal{E}_{3}:T_{3}=0$ (in dashed line) and $\mathcal{E}_{1}:T_{1}=0$ (in
dotted line) are ellipses;

– $\mathcal{H}:T_{4}=0$ (in solid line) is a hyperbola;

– $\mathcal{P}:C=B^{2}/4$ is a parabola (in dash-dotted line).

Figure 1. The set $A^{\dagger}_{0}(\sigma_{\bullet})$ subdued to the condition
$f_{2}=f_{3}$ (global view).)

######  Remark 2.

As $C>0$, only the branch of $\mathcal{H}$ belonging to the upper half-plane
is represented in Fig. 1 and 2. The asymptotes of $\mathcal{H}$ are the lines
$B=-1$ and $C=3B-6$. We denote by Int$(\mathcal{E}_{i})$ and
Out$(\mathcal{E}_{i})$ the intersections with the half-plane $C>0$ of the
interior and the exterior of the ellipse $\mathcal{E}_{i}$. By
Int$(\mathcal{H})$ we denote the part of the upper half-plane which is above
and by Out$(\mathcal{H})$ the part which is below the branch of $\mathcal{H}$
with $C>0$. Notice that

$\begin{array}[]{lll}{\rm Int}(\mathcal{E}_{3}):~{}T_{3}>0,~{}C>0,&{\rm
Int}(\mathcal{E}_{1}):~{}T_{1}<0,~{}C>0,&{\rm
Int}(\mathcal{H}):~{}T_{4}<0,~{}C>0,\\\ \\\ {\rm
Out}(\mathcal{E}_{3}):~{}T_{3}<0,~{}C>0,&{\rm
Out}(\mathcal{E}_{1}):~{}T_{1}>0,~{}C>0,&{\rm
Out}(\mathcal{H}):~{}T_{4}>0,~{}C>0.\end{array}$

The ellipse $\mathcal{E}_{1}$ intersects the $C$-axis at $(0,0)$ and $(0,1/5)$
while $\mathcal{E}_{3}$ is tangent to the $C$-axis at $(0,1)$. The leftmost
point of the ellipse $\mathcal{E}_{1}$ is at

$((8-\sqrt{70})/12=-0.030\ldots,(10-\sqrt{70})/20=0.081\ldots)~{}.$

The point $(4/3,1)$ is a common point for $\mathcal{L}$, $\mathcal{L}_{0}$,
$\mathcal{H}$, $\mathcal{E}_{1}$ and $\mathcal{E}_{3}$.

The intersecting lines $\mathcal{L}_{1}$ and $\mathcal{L}$ define two pairs of
opposite sectors. The ones of opening $>\pi/2$ are denoted by
$\mathcal{S}_{u}:T_{0}<0,D<0$ (upper) and $\mathcal{S}_{\ell}:T_{0}>0,D>0$
(lower). One has $A>0$ exactly when the point $(B,C)$ belongs to one of these
two sectors.

Figure 2. The set $A^{\dagger}_{0}(\sigma_{\bullet})$ subdued to the condition
$f_{2}=f_{3}$ (local view).)

I) The signs of the coefficients $f_{i}$ and of the quantities $A>0$ and $C>0$
imply that one must have one of the two systems of conditions:

$\begin{array}[]{l}{\rm(i)}~{}:~{}(B,C)\in\mathcal{S}_{\ell}\cap{\rm
Int}(\mathcal{E}_{1})\cap{\rm Int}(\mathcal{E}_{3})\cap{\rm
Int}(\mathcal{H})~{},~{}~{}~{}\,\,{\rm i.e.}\\\ \\\
~{}~{}~{}~{}~{}\,\,T_{0}>0~{},~{}~{}~{}D>0~{},~{}~{}~{}T_{1}<0~{},~{}~{}~{}T_{3}>0~{}~{}~{}\,\,{\rm
and}~{}~{}~{}\,\,T_{4}<0~{}~{}~{}\,\,{\rm or}\\\ \\\
{\rm(ii)}~{}:~{}(B,C)\in\mathcal{S}_{u}\cap{\rm Out}(\mathcal{E}_{1})\cap{\rm
Out}(\mathcal{E}_{3})\cap{\rm Out}(\mathcal{H})~{},~{}~{}~{}\,\,{\rm i.e.}\\\
\\\
~{}~{}~{}~{}~{}\,\,T_{0}<0~{},~{}~{}~{}D<0~{},~{}~{}~{}T_{1}>0~{},~{}~{}~{}T_{3}<0~{}~{}~{}\,\,{\rm
and}~{}~{}~{}\,\,T_{4}>0~{}.\end{array}$

The possibility (i) is to be excluded. Indeed, one has

$\mathcal{E}_{3}\cap\mathcal{L}_{0}=\\{(2/3,1/3)~{},~{}~{}~{}(4/3,1)\\}~{}~{}~{}\,\,{\rm
and}~{}~{}~{}\,\,\mathcal{E}_{3}\cap\mathcal{L}=\\{(4/3,1)~{},~{}~{}~{}(2,3)\\}~{},$

see Fig. 1 and 2, so Int$(\mathcal{E}_{3})$ intersects with $\mathcal{S}_{u}$,
but not with $\mathcal{S}_{\ell}$.

J) We describe the set obtained in case (ii). For $B\leq-1$, this is the part
of the upper plane which is above the parabola $\mathcal{P}$. For
$-1<B<(8-\sqrt{70})/12$, this is its part between the parabola $\mathcal{P}$
from below and the hyperbola $\mathcal{H}$ from above, see Fig. 1. For each
$(8-\sqrt{70})/12\leq B<0$, this is the union of two intervals whose endpoints
belong to $\mathcal{H}$ and $\mathcal{E}_{1}$ for the upper and to
$\mathcal{E}_{1}$ and $\mathcal{P}$ for the lower interval. For $B\geq 0$,
this is the union of two curvilinear triangles, each with one rectilinear side
which is part of the $C$-axis. The above triangle has vertices at $(0,1)$,
$(0,5)$ and $(0.34\ldots,2.42\ldots)$. The latter point, together with
$(4/3,1)$, is the intersection $\mathcal{H}\cap\mathcal{E}_{3}$.

The lower triangle has vertices at $(0,1/5)$, $(0,1)$ and
$(0.14\ldots,0.41\ldots)$. The latter point, together with $(4/3,1)$, is the
intersection $\mathcal{E}_{1}\cap\mathcal{E}_{3}$.

To see that there is no other point of the set defined in case (ii) with
$B>0$, one has to observe the order on $\mathcal{P}$ of the intersection
points of

$\begin{array}[]{l}\mathcal{P}\cap\mathcal{L}_{0}=\\{(0.36\ldots,~{}0.03\ldots),~{}(3.63\ldots,~{}3.29\ldots)\\}~{}~{}~{}\,\,{\rm
and}\\\ \\\
\mathcal{P}\cap\mathcal{E}_{1}=\\{(0,~{}0),~{}(0.47\ldots,~{}0.22\ldots)\\}~{}.\end{array}$

The connectedness of the set obtained in case (ii) follows from its
description. ∎

##  4\. Proof of Theorem 3

Part (1). The last component of $\sigma_{\triangle}$ is a $+$. Suppose that
there is a minus sign in $\sigma_{\triangle}$ corresponding to $x^{2m}$,
$1\leq m\leq[d/2]$. The polynomial $-x^{2m}+1$ has exactly two real roots,
namely $\pm 1$, and they are simple. For $\varepsilon>0$ small enough, the
polynomial $P_{0}:=\varepsilon x^{d}-x^{2m}+1$ has exactly three real roots
two of which are close to $\pm 1$ and the third is $>1$. (One can notice that
by Descartes’ rule of signs it has not more than two positive and not more
than one negative root.)

Fix a degree $d$ polynomial $P_{1}$ with $\sigma(P_{1})=\sigma_{\triangle}$.
Then for $0<\eta\ll\varepsilon$, the polynomial $P_{0}+\eta P_{1}$ has signs
of the coefficients as defined by $\sigma_{\triangle}$ and has exactly one
negative and two positive simple roots and $(d-3)/2$ complex conjugate pairs
counted with multiplicity. Thus $P_{0}+\eta P_{1}$ realizes the couple
$(\sigma_{\triangle},(2,1))$.

Suppose now that there are $(+)$-signs in $\sigma_{\triangle}$ corresponding
to all monomials of even degrees. Then there is a monomial $x^{2m+1}$, $1\leq
2m+1<d$, whose sign is negative. The polynomial $P_{2}:=x^{d}-x^{2m+1}$ has
simple roots at $\pm 1$ and a $(2m+1)$-fold root at $0$. For $\varepsilon>0$
small enough, the polynomial $P_{2}+\varepsilon$ has exactly three real roots
(two positive and one negative) all of which are simple. Then with $P_{1}$ and
$\eta$ as above, the polynomial $P_{2}+\varepsilon+\eta P_{1}$ realizes the
couple $(\sigma_{\triangle},(2,1))$.

Part (2). We construct a polynomial of the form
$V:=x^{d}-Ax^{2m}-Bx^{2n-1}+C$, $A>0$, $B>0$, $C>0$, such that
$V(1)=V^{\prime}(1)=V(-1)=0$:

(4.4) $\begin{array}[]{lll}1-A-B+C=0~{},&-1-A+B+C=0~{},&d-2mA-(2n-1)B=0\\\ \\\
{\rm hence}&A=C=(d-2n+1)/2m~{},&B=1~{}.\end{array}$

By Descartes’ rule of signs, $V$ has no other real roots. After this one
decreases $C$: $C\mapsto C-t$, $t\geq 0$. For $t=0$, the root $-1$ moves with
a finite speed to the right while the double root at $1$ splits into two real
roots moving for $t=0$ with infinite speeds to the left and right
respectively. Hence for $t>0$ close to $0$, one has
$\alpha_{1}<\beta<\alpha_{2}$. The linear system (4.4) with unknown variables
$A$, $B$ and $C$ has non-zero determinant. Hence for $\varepsilon>0$ small
enough, one can obtain polynomials $V$ satisfying the conditions

$V(1)=V^{\prime}(1)=V(-1\pm\varepsilon)=0~{}~{}~{}\,\,{\rm(resp.}~{}~{}~{}\,\,V(1)=V^{\prime}(1)\pm\varepsilon=V(-1)=0{\rm)}$

which after decreasing $C$ yield polynomials satisfying the inequalities
$\beta<\alpha_{1}<\alpha_{2}$ or $\alpha_{1}<\alpha_{2}<\beta$ (resp. the
conditions $\beta=\alpha_{1}<\alpha_{2}$ or $\alpha_{1}<\alpha_{2}=\beta$). It
remains to construct the polynomial $V+\eta P_{1}$, where
$0<\eta\ll\varepsilon$ and $\sigma(P_{1})=\sigma_{\triangle}$.

Part (3). There exists a monomial $x^{2m}$ with negative coefficient. Then for
$\varepsilon>0$ small enough, the polynomial
$W:=x^{2m-1}(x-1)(x-2)+\varepsilon$ has exactly one negative and two positive
roots whose moduli satisfy the condition $\beta<\alpha_{1}<\alpha_{2}$. Its
four non-zero coefficients have the signs as defined by $\sigma_{\triangle}$.
After this one constructs the polynomial $W+\eta P_{1}$ with $\eta$ and
$P_{1}$ as above.

The inequality $\beta\geq\alpha_{1}$ is impossible. Indeed, represent a
polynomial $W$ realizing the couple $\mathcal{C}$ in the form $W=W_{o}+W_{e}$,
where $W_{o}$ is the odd and $W_{e}$ is the even part of $W$. Then for
$x\in(-\beta,0)$, one has $W_{e}(x)=W_{e}(-x)$ and $W_{o}(x)<W_{o}(-x)$. As
$W(x)>0$ for $x\in(-\beta,0)$, one cannot have $W(\alpha_{1})=0$. This is a
contradiction.

Part (4). Changing the polynomial $Y(x)$ with $\sigma(Y)=\sigma_{\triangle}$
which realizes the couple $\mathcal{C}$ to $Y_{1}:=x^{d}Y(1/x)/Y(0)$ (we set
$\sigma_{\triangle}^{R}:=\sigma(Y_{1})$) one obtains a polynomial realizing
the couple $(\sigma_{\triangle}^{R},(2,1))$, where all odd monomials have
positive signs, see Definition 4. The roots of $Y_{1}$ are the reciprocals of
the roots of $Y$, so one deduces part (4) from part (3).

##  5\. Proof of Theorem 4

The last sign of $\sigma_{\diamond}$ is a $-$. Suppose that there are two
monomials $x^{2m}$ and $x^{2p}$, $m>p>0$, whose signs defined by
$\sigma_{\diamond}$ are $-$ and $+$ respectively. Consider the polynomial
$P_{3}:=-x^{2m}+Ax^{2p}-B$, $A>0$, $B>0$. By Descartes’ rule of signs it has
at most two positive and at most two negative roots. We define $A$ and $B$
such that $P_{3}$ has double roots at $1$ and $(-1)$:

$\begin{array}[]{lll}-1+A-B=0~{},&-2m+2pA=0&{\rm hence}\\\ \\\
A=m/p>0~{},&B=(m-p)/p>0~{}.&\end{array}$

Then for $\varepsilon>0$ small enough, the polynomial $P_{3}+\varepsilon
x^{d}$ has exactly three real roots, all simple and positive. Suppose that
$P_{4}$ is a degree $d$ polynomial such that
$\sigma(P_{4})=\sigma_{\diamond}$. Then for $0<\eta\ll\varepsilon$, the
polynomial $P_{3}+\varepsilon x^{d}+\eta P_{4}$ has sign pattern
$\sigma_{\diamond}$ and has exactly three real roots, all simple and positive.

Suppose that there are no monomials $x^{2m}$ and $x^{2p}$ as above. Then the
signs of the first $a$ even monomials are positive and the ones of the last
$(d+1-2a)/2$ of them are negative, $0\leq a\leq(d-1)/2$. Suppose that there
are monomials $x^{2\nu}$, $x^{2\mu-1}$ and $x^{2\theta}$,
$2\nu>2\mu-1>2\theta$, whose signs defined by $\sigma_{\diamond}$ are $-$, $+$
and $-$ respectively. By Descartes’ rule of signs a polynomial of the form
$P_{5}:=-x^{2\nu}+Cx^{2\mu-1}-Dx^{2\theta}$, $C>0$, $D>0$, has at most two
positive roots and no negative roots; clearly it has a $(2\theta)$-fold root
at $0$. One can choose $C$ and $D$ such that the positive roots are at $1$ and
$2$:

$\begin{array}[]{lll}-1+C-D=0~{},&-2^{2\nu}+2^{2\mu-1}C-2^{2\theta}D=0&{\rm
hence}\\\ \\\
D=(2^{2\nu}-2^{2\mu-1})/(2^{2\mu-1}-2^{2\theta})>0~{},&C=D+1>0~{}.&\end{array}$

For $\varepsilon>0$ small enough, the polynomial $P_{5}+\varepsilon x^{d}$ has
three positive simple roots and no other real roots, and the polynomial
$P_{6}:=P_{5}+\varepsilon x^{d}+\eta P_{4}$ with $\eta$ and $P_{4}$ as above
has three positive simple roots, no other real roots and
$\sigma(P_{6})=\sigma_{\diamond}$.

So now we suppose that there are no monomials $x^{2m}$ and $x^{2p}$, and no
monomials $x^{2\nu}$, $x^{2\mu-1}$ and $x^{2\theta}$ as above. Suppose that
there are monomials $x^{2u-1}$ and $x^{2v-1}$, $d>2u-1>2v-1>0$, such that
their signs are $-$ and $+$ respectively. One can construct a polynomial
$P_{7}:=x^{d}-Ex^{2u-1}+Fx^{2v-1}$, $E>0$, $F>0$, having double roots at $\pm
1$, a $(2v-1)$-fold root at $0$ and no other real roots:

$\begin{array}[]{lll}1-E+F=0~{},&d-(2u-1)E+(2v-1)F=0&{\rm hence}\\\ \\\
F=(d-2u+1)/2(u-v)>0~{},&E=F+1>0~{}.&\end{array}$

The absence of other real roots is guaranteed by Descartes’ rule of signs.
Hence for $0<\eta\ll\varepsilon\ll 1$, the polynomial $P_{7}-\varepsilon+\eta
P_{4}$ has sign pattern $\sigma_{\diamond}$, three simple positive roots and
no other real roots (recall that $P_{4}(0)<0$).

Suppose that there are no couples or triples of monomials $x^{2m}$, $x^{2p}$
or $x^{2\nu}$, $x^{2\mu-1}$, $x^{2\theta}$ or $x^{2u-1}$, $x^{2v-1}$. Then the
signs of the first $h_{o}\geq 1$ odd monomials (including $x^{d}$) are
positive and the signs of the remaining $(d+1-2h_{o})/2$ odd monomials are
negative. The signs of the first $h_{e}\geq 0$ even monomials are positive and
the signs of the other $(d+1-2h_{e})/2$ ones are negative. The absence of
triples $x^{2\nu}$, $x^{2\mu-1}$, $x^{2\theta}$ implies $h_{o}\leq h_{e}+1$.
The cases $h_{o}=h_{e}+1$ and $h_{o}=h_{e}$ are impossible, because there is
only one sign change in the sign pattern. Therefore $1\leq h_{o}\leq h_{e}-1$.
This means that the sign pattern is $D(a,b,c)$ with $a=h_{o}$, $b=h_{e}-h_{o}$
and $c=(d+1-2a-2b)/2$.

##  6\. Proof of Theorem 5

Suppose that a polynomial $P:=\sum_{j=0}^{d}a_{j}x^{j}$ realizes the couple
$(D(a,b,c),(3,0))$. Denote by

$P_{o}:=\sum_{\nu=0}^{(d-1)/2}a_{2\nu+1}x^{2\nu+1}~{}~{}~{}{\rm
and}~{}~{}~{}P_{e}:=\sum_{\nu=0}^{(d-1)/2}a_{2\nu}x^{2\nu}$

its odd and even parts respectively. In each of the sequences
$\\{a_{2\nu+1}\\}_{\nu=0}^{(d-1)/2}$ and $\\{a_{2\nu}\\}_{\nu=0}^{(d-1)/2}$
there is exactly one sign change. Descartes’ rule of signs implies that the
polynomial $P_{o}$ has exactly three real roots, namely $-x_{o}$, $0$ and
$x_{o}$, $x_{o}>0$, while the polynomial $P_{e}$ has exactly two real roots
$\pm x_{e}$, $x_{e}>0$; all these five roots are simple.

######  Remarks 2.

(1) The polynomial $P_{e}$ is positive and increasing on $(x_{e},\infty)$ and
negative on $[0,x_{e})$. The polynomial $P_{o}$ is positive and increasing on
$(x_{o},\infty)$ and negative on $(0,x_{o})$.

(2) One has $x_{o}\neq x_{e}$, otherwise $P(-x_{o})=0$, i.e. $P$ has a
negative root which is a contradiction.

(3) One can assume that all positive roots of $P$ are distinct. Indeed, if
this is not the case, then one can perturb $P$ to make all its positive roots
distinct without changing the signs of its coefficients as follows. If $P$ has
an $\ell$-fold root $\lambda>0$ ($\ell>1$), i.e. $P=(x-\lambda)^{\ell}P^{0}$,
$P^{0}(\lambda)\neq 0$, then for $\varepsilon>0$ small enough, the polynomial
$(x-\lambda)^{\ell-1}(x-\lambda-\varepsilon)P^{0}$ has the same sign pattern
and its $\ell$-fold root has split into an $(\ell-1)$-fold and a simple real
roots. It remains to iterate this construction sufficiently many times.

######  Notation 3.

We denote by $0<\xi_{1}<\xi_{2}<\xi_{3}$ the smallest three of the positive
roots of $P$ and by $\zeta$ a positive number different from $x_{o}$ and
$x_{e}$.

It is clear that $P(\zeta)>0$ for $\zeta\in(\xi_{1},\xi_{2})$ and $P(\zeta)<0$
for $\zeta\in(\xi_{2},\xi_{3})$. For $\zeta\in(\xi_{1},\xi_{2})$, it is
impossible to have $P_{e}(\zeta)\leq 0$ and $P_{o}(\zeta)\leq 0$ (with at most
one equality, see part (2) of Remarks 2). It is also impossible to have
$P_{e}(\zeta)\geq 0$ and $P_{o}(\zeta)\geq 0$. Indeed, this would imply that
$x_{e}\leq\zeta<\xi_{2}$ and $x_{o}\leq\zeta<\xi_{2}$ which means that for
$x\in(\xi_{2},\xi_{3})$, one has $P_{e}(x)\geq 0$ and $P_{o}(x)\geq 0$, i.e.
$P(x)>0$. This is a contradiction.

Two possible situations are left:

a) $P_{e}(\zeta)>0$, $P_{o}(\zeta)<0$;

b) $P_{e}(\zeta)<0$, $P_{o}(\zeta)>0$

(we skip the cases of equalities, because they were already taken into
account).

Situation a) cannot take place, because this would mean that

$P(-\zeta)=P_{e}(\zeta)-P_{o}(\zeta)>0~{},$

and since $P(0)<0$ and $P(x)\rightarrow-\infty$ for $x\rightarrow-\infty$, in
each of the intervals $(-\infty,-\zeta)$ and $(-\zeta,0)$ the polynomial $P$
would have at least one root – a contradiction.

So suppose that we are in situation b), so $x_{o}<\zeta<x_{e}$. Without loss
of generality one can assume that $\xi_{1}=1$; this can be achieved by a
rescaling $x\mapsto\xi_{1}x$. Hence $P_{o}(1)=\beta>0$ and $P_{e}(1)=-\beta$.
Considering the polynomial $P/\beta$ instead of $P,$ one can assume that
$\beta=1$. One deduces from Lemma 1 which follows that there are no real roots
of $P$ larger than $1$ (one can use the Taylor series of $P$ at $1$); this
contradiction completes the proof.

######  Lemma 1.

Under the above assumptions, $P^{(m)}(1)>0$, for any $m=1,2,\ldots,d$.

###### Proof of Lemma 1..

In the proof we allow zero values of the coefficients as well. This is because
we need to deal with compact sets on which minimization arguments are to be
applied.

Suppose that the sum $\delta_{1}:=a_{1}+a_{3}+\cdots+a_{2b+2c-1}$ is fixed
(recall that these are all the negative coefficients of $P_{o}$). Then for any
$m=1$, $2$, $\ldots$, $d,$ it is true that $P_{o}^{(m)}(1)$ is minimal for

$a_{2b+2c-1}=\delta_{1}~{},~{}~{}~{}\,a_{1}=a_{3}=\cdots=a_{2b+2c-3}=0~{}.$

Indeed, when computing the values of the derivatives at $x=1$, monomials of
larger degree in $x$ are multiplied by larger factors (equal to these
degrees). We apply here $(d-3)/2$ times the fact that for $A+B$ fixed, the
inequalities $A\geq 0$, $B\geq 0$ and $\lambda>\mu>0$ imply that the sum
$\lambda A+\mu B$ is maximal when $B=0$.

Similarly, if the sum $\delta_{2}:=a_{2b+2c+1}+a_{2b+2c+3}+\cdots+a_{d}$ of
all positive coefficients of $P_{o}$ is fixed, then $P_{o}^{(m)}(1)$ is
minimal for $a_{2b+2c+1}=\delta_{2}$, $a_{2b+2c+3}=\cdots=a_{d}=0$.

For the polynomial $P_{e}$ we obtain in the same way that if the sums

$\delta_{3}:=a_{0}+a_{2}+\cdots+a_{2c-2}~{}~{}~{}\,{\rm
and}~{}~{}~{}\,\delta_{4}:=a_{2c}+\cdots+a_{d-1}$

are fixed, then $P_{e}^{(m)}(1)$ is minimal for $a_{2c-2}=\delta_{3}$,
$a_{0}=a_{2}=\cdots=a_{2c-4}=0$, $a_{2c}=\delta_{4}$,
$a_{2c+2}=\cdots=a_{d-1}=0$. Thus the polynomials $P_{o}$ and $P_{e}$ are of
the form

$P_{o}=Ex^{2b+2c+1}-Fx^{2b+2c-1}~{}~{}~{},~{}~{}~{}P_{e}=Gx^{2c}-Hx^{2c-2}~{},$

with $E:=a_{2b+2c+1}\geq 0$, $-F:=a_{2b+2c-1}\leq 0$, $G:=a_{2c}\geq 0$ and
$-H:=a_{2c-2}\leq 0$. Recall that

$P(1)=0~{},~{}~{}~{}P_{o}(1)=1~{}~{}~{}{\rm
and}~{}~{}~{}P_{e}(1)=-1~{},~{}~{}~{}{\rm i.~{}e.}~{}~{}~{}E-F=1~{}~{}~{}{\rm
and}~{}~{}~{}G-H=-1~{}.$

The values of the derivatives at $x=1$ are of the form

$P^{(m)}(1)=u_{m}E-v_{m}F+w_{m}G-t_{m}H~{},~{}~{}~{}\,\,u_{m}>v_{m}>w_{m}>t_{m}~{},$

with $u_{m},~{}v_{m},w_{m},~{}t_{m}\in\mathbb{N}$. Hence

$\begin{array}[]{lll}P^{(m)}(1)&=&(u_{m}-v_{m})E+v_{m}(E-F)+(w_{m}-t_{m})G+t_{m}(G-H)\\\
\\\ &=&(u_{m}-v_{m})E+(w_{m}-t_{m})G+(v_{m}-t_{m})>0~{}.\end{array}$

∎

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