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##  Appendix A Specific angular momentum at the innermost stable circular
orbit

The radius of the innermost stable circular orbit (ISCO) is given by
$R_{\mathrm{ISCO}}=r_{\mathrm{ISCO}}R_{\mathrm{G}}$, where
$R_{\mathrm{G}}=M_{\mathrm{BH}}G/c^{2}$ and $r_{\mathrm{ISCO}}$ is the
dimensionless radius of the ISCO. $r_{\mathrm{ISCO}}$ depends on spin as

$r_{\mathrm{ISCO}}=3+Z_{2}\mp\sqrt{(3-Z_{1})(3+Z_{1}+2Z_{2})},$ (13)

where the minus and plus sign are for prograde and retrograde accretion,
respectively, and $Z_{1}$ and $Z_{2}$ are functions of spin given by

$Z_{1}(a)=1+\Big{(}1-a^{2}\Big{)}^{1/3}\Big{[}\big{(}1+|a|\big{)}^{1/3}+\big{(}1-|a|\big{)}^{1/3}\Big{]}$
(14)

and

$Z_{2}(a)=\sqrt{3a^{2}+Z_{1}}.$ (15)

The specific angular momentum at the ISCO is given by
$L_{\mathrm{ISCO}}=M_{\mathrm{BH}}G\ell_{\mathrm{ISCO}}/c$, where
$\ell_{\mathrm{ISCO}}$ is a dimensionless function of spin:

$\ell_{\mathrm{ISCO}}(a)=\frac{2}{3\sqrt{3}}\Big{(}1+2\sqrt{3r_{\mathrm{ISCO}}-2}\Big{)}.$
(16)

##  Appendix B Black hole spin alignment timescale

Our procedure for modeling SMBH spin builds on the approach of several
previous studies (Volonteri et al. 2007, King et al. 2008, Fanidakis et al.
2011, Griffin et al. 2019), with the main difference being that those studies
focused on thin, radiatively efficient discs. This method is similar in many
aspects to that implemented by Fiacconi et al. (2018) and Talbot et al.
(2020), which also modeled the thin disc. However, we do not explicitly model
the differential equation for the evolution of the SMBH spin direction due to
LT torques between the accretion disk and the SMBH (e.g. Martin et al. 2007).

Furthermore, Fiacconi et al. (2018) measure the inflow of mass and angular
momentum onto their sink particle, representing the SMBH and accretion disc
system (without assuming e.g. Bondi accretion). This allows them to model the
size, as well as total mass and angular momentum of their subgrid accretion
disc (assuming some surface density profile, as well as Keplerian rotation).
In turn this leads to the mass and angular momentum accretion rates onto the
SMBH, using a system of equations that couple the SMBH with the large-scale
accretion disc. These accretion rates onto the SMBH are in general different
from the inflow rates onto the sink particle.

We instead assume Bondi accretion, and that the accretion rate onto the SMBH
is equal to the Bondi rate at all times. For the angular momentum evolution,
we use only the direction of angular momentum measured in the smoothing kernel
of the SMBH. While it is beyond the scope of this paper to compare our
approach with the full approach of Fiacconi et al. (2018), we will compare the
alignment timescale in our approach to that where LT torques are explicitly
included in the angular momentum evolution equation. We do this for the
radiatively-efficient thin disc assuming that gas pressure dominates over
radiation pressure, and that free-free absorption dominates in the opacity
(region C from Shakura & Sunyaev 1973). We make this choice since Fiacconi et
al. (2018) studied this accretion regime, and since they use the derivation of
Martin et al. (2007) to model the terms from the LT torque that contribute to
the angular momentum evolution equation, which is applicable only for
Keplerian rotation and thin discs. Note that while we assume the thick
accretion disc in this paper, a comparison between the differential equation
approach and the incremental accretion approach in any accretion regime should
serve as a general validation of the incremental accretion approach (with the
caveat that the differential equation itself should change depending on
accretion regime).

In the Fiacconi et al. (2018) approach, the evolution of angular momentum is
given by the equation

$\displaystyle\frac{\mathrm{d}\mathbf{J}_{\mathrm{BH}}}{\mathrm{d}t}=$
$\displaystyle
L_{\mathrm{ISCO}}\dot{M}_{\mathrm{BH,0}}\mathbf{\hat{J}_{\mathrm{BH}}}-$ (17)
$\displaystyle\frac{J_{\mathrm{BH}}}{\tau_{\mathrm{GM}}}\big{[}\tilde{K}_{1}\mathbf{\hat{J}_{\mathrm{BH}}}\times\mathbf{\hat{J}_{\mathrm{d}}}+\tilde{K}_{2}\mathbf{\hat{J}_{\mathrm{BH}}}\times(\mathbf{\hat{J}_{\mathrm{BH}}}\times\mathbf{\hat{J}_{\mathrm{d}}})\big{]},$

where $\mathbf{J}_{\mathrm{BH}}$ is the angular momentum vector of the BH,
$\mathbf{\hat{J}_{\mathrm{BH}}}$ is its direction,
$\mathbf{\hat{J}_{\mathrm{d}}}$ is the direction of the large-scale accretion
disc surrounding the black hole (outside the warp radius),
$\tilde{K}_{1}=\sin(\pi/7)$, $\tilde{K}_{2}=\cos(\pi/7)$ and
$\tau_{\mathrm{GM}}$ is a gravito-magnetic timescale which is given by

$\tau_{\mathrm{GM}}=0.17\hskip 0.85358pt\mathrm{Myr}\hskip
0.85358pt\bigg{(}\frac{M_{\mathrm{BH}}}{10^{6}\hskip
0.85358pt\mathrm{M}_{\odot}}\bigg{)}^{-2/35}\dot{m}^{-32/35}|a|^{5/7}.$ (18)

for the thin accretion disc. The first term in equation (17) corresponds to
accretion onto the SMBH, and is identical to what we assume. Other terms
represent the effects of LT torques in a warped accretion disc. Their form,
including the forms of $\tilde{K}_{1}$, $\tilde{K}_{2}$ and
$\tau_{\mathrm{GM}}$, follow from the derivation in Martin et al. (2007) under
the assumption that the surface density scales as $\Sigma(R)\sim R^{-3/4}$, as
appropriate for region C of the Shakura & Sunyaev (1973) thin disc, and that
the tilt angle between the SMBH spin vector and the outer accretion disc is
small.

The first term in the parentheses in equation (17) causes precession, whereas
the second term leads to alignment. If we define the alignment timescale such
that the alignment term in equation (17) takes the form
$(\mathrm{d}\mathbf{J}_{\mathrm{BH}}/\mathrm{d}t)_{\mathrm{align}}=-(J_{\mathrm{BH}}/\tau_{\mathrm{align}})\mathbf{\hat{J}_{\mathrm{BH}}}\times(\mathbf{\hat{J}_{\mathrm{BH}}}\times\mathbf{\hat{J}_{\mathrm{d}}})$,
the alignment timescale in this differential equation approach can be written
as

$\displaystyle\tau_{\mathrm{align,diff.\hskip 0.42677pteqn.}}$
$\displaystyle=\frac{\tau_{\mathrm{GM}}}{\cos(\pi/7)}=$ (19)
$\displaystyle=0.19\hskip 0.85358pt\mathrm{Myr}\hskip
0.85358pt\bigg{(}\frac{M_{\mathrm{BH}}}{10^{6}\hskip
0.85358pt\mathrm{M}_{\odot}}\bigg{)}^{-2/35}\dot{m}^{-32/35}|a|^{5/7}.$

In the approach we use in this paper (we refer to this as the warp increment
approach), we use only the first term of equation (17) to evolve the magnitude
of angular momentum (i.e. spin). To evolve its direction, at the end of every
time-step we assume that the new direction of angular momentum matches that of
$\mathbf{J}_{\mathrm{BH}}+\Delta\mathbf{J}_{\mathrm{warp,tot}}$, where
$\Delta\mathbf{J}_{\mathrm{warp,tot}}=\Delta
J_{\mathrm{warp,tot}}\mathbf{\hat{J}}_{\mathrm{d}}$ is the total angular
momentum of all of the warp increments consumed over the time-step. This
material is originally aligned with the outer accretion disc (in the direction
of $\mathbf{\hat{J}}_{\mathrm{d}}$), but we assume that it is
(counter-)aligned with the SMBH angular momentum vector through LT torques. In
the process, the SMBH is also torqued in the opposite direction. The magnitude
of the consumed angular momentum is given by $\Delta
J_{\mathrm{warp,tot}}=N_{\mathrm{warp}}J_{\mathrm{warp}}$, where
$N_{\mathrm{warp}}=\Delta M_{0}/M_{\mathrm{warp}}$ is the number of warp
increments consumed over the time-step, given the mass to be consumed, $\Delta
M_{0}$, and $J_{\mathrm{warp}}$ is the angular momentum of one warp increment.
Equally, one can view this as the SMBH being torqued by gas with a specific
angular momentum of $L_{\mathrm{warp}}=J_{\mathrm{warp}}/M_{\mathrm{warp}}$ as
it accretes. For the general case of $\Sigma(R)\sim R^{p}$, we have
$L_{\mathrm{warp}}=(2+p)/(5/2+p)\times\sqrt{M_{\mathrm{BH}}GR_{\mathrm{warp}}}$,
where $R_{\mathrm{warp}}$ is the warp radius (see appendix in Fiacconi et al.
2018 for expression for region C of the thin disc). For the case of interest
here, $\Sigma\sim R^{-3/4}$, and thus the numerical factor evaluates to $5/7$.
Using the above relations, we can define a similar alignment timescale to the
one in the differential equation approach, namely

$\displaystyle\tau_{\mathrm{align,warp}}$
$\displaystyle=\bigg{(}\frac{J_{\mathrm{BH}}\Delta t}{\Delta
J_{\mathrm{warp,tot}}}\bigg{)}=$ (20) $\displaystyle=0.21\hskip
0.85358pt\mathrm{Myr}\hskip
0.85358pt\bigg{(}\frac{M_{\mathrm{BH}}}{10^{6}\hskip
0.85358pt\mathrm{M}_{\odot}}\bigg{)}^{-2/35}\dot{m}^{-32/35}|a|^{5/7}.$

This alignment timescale differs from the differential equation one by an
order-unity numerical factor. However, note that they are only directly
comparable in the case that the SMBH angular momentum vector, and the angular
momentum vector of the outer accretion disc, are perpendicular. Furthermore,
we generally do not expect the two timescales to be the same, since these two
approaches appear inherently different.

In order to compare the two approaches in more detail, we use a very simple
test set-up. We assume a SMBH mass of $M_{\mathrm{BH}}=10^{7}$
$\mathrm{M}_{\odot}$, spin of $a=0.5$ directed along the $z-$axis, and
constant accretion rate of $\dot{m}=0.1$. We assume that the angular momentum
of the accretion disc on large scales, $\mathbf{\hat{J}}_{\mathrm{d}}$, is
directed along the $x-$axis. Our aim is to compare how long it takes for the
spin vector to be redirected in the direction of
$\mathbf{\hat{J}}_{\mathrm{d}}$ in the two approaches. With our assumed
parameters, we find that both the mass and spin magnitude remain very close to
their initial values over the timescale of realignment, which is in this case
of order a few Myr. We evolve the system using $100$ time steps between $t=0$
Myr and $t=10$ Myr in both approaches.

In Fig. 11 we show the $x-$component of the angular momentum direction of the
SMBH, i.e. the component in the direction of the outer accretion disc. The
alignment is complete within $2-3$ Myr in both approaches, but it appears to
be somewhat slower in our approach, using warp increments. We have attempted
our approach with the warp angular moomentum boosted slightly so that the
alignment time-scales is exactly equal to the one in the differential equation
approach (i.e. so that the numerical factor in equation 20 evaluates to 0.19
instead of 0.21, to match equation 19). This change leads to the two
approaches of modeling SMBH spin alignment agreeing perfectly, showing that
they are compatible.

Since we assume a thick disc in the model presented in this paper, rather than
the thin disc, it is worth addressing the alignment timescale for that
accretion regime. The surface density in the self-similar thick solution of
Narayan & Yi (1995) scales as $\Sigma(R)\sim R^{-1/2}$, so
$L_{\mathrm{warp}}=(3/4)\Omega_{0}\sqrt{M_{\mathrm{BH}}GR_{\mathrm{warp}}}$,
where we have also assumed that the orbital velocities are a fraction
$\Omega_{0}$ of their Keplerian values (see discussion in §2.6). Defining the
alignment timescale in the same way as in equation (20) leads to

$\tau_{\mathrm{align,warp,thick}}=\frac{4M_{\mathrm{BH}}}{3\dot{M}_{\mathrm{BH,0}}}|a|\Omega_{0}\bigg{(}\frac{384|a|}{25(H/R)^{2}}\bigg{)}^{-1/5},$
(21)

where we have used equation (8) for the warp radius of the thick disc. Using
our fiducial values for the parameters leads to
$\tau_{\mathrm{align,warp,thick}}=0.63|a|^{4/5}(M_{\mathrm{BH}}/\dot{M}_{\mathrm{BH,0}})$.
This shows that the alignment timescale of a thick disc is of order the growth
timescale of the SMBH. This is generally much slower than in the thin disc
case, owing to the vast difference in the warp radii (several vs. thousands of
$R_{\mathrm{G}}$, respectively), and therefore even larger differences in warp
angular momenta. For a value of $a=0.25$ (the equilibrium spin value, relevant
in our high-mass galaxy cluster simulations), the alignment timescale is
significantly shorter than the growth timescale,
$\tau_{\mathrm{align,warp,thick}}\approx
0.2(M_{\mathrm{BH}}/\dot{M}_{\mathrm{BH,0}})$, indicating that the SMBH can be
redirected without having to significantly grow its mass.

Figure 11: The alignment of an initially misaligned SMBH spin vector,
calculated using two different approaches. We show the $x-$ component of the
angular momentum direction vector, i.e. the component in the direction of the
outer accretion disc. The initial SMBH spin is $a=0.5$, and it is directed
along the $z-$axis. The SMBH mass is $M_{\mathrm{BH}}=10^{7}$
$\mathrm{M}_{\odot}$, and the SMBH is accreting with a constant $\dot{m}=0.1$.
Black lines show the predicted alignment from the differential equation
approach, using equations (17) and (19), while blue lines show predictions
using our warp increment approach, with equation (20). The solid blue line
shows the predicted alignment using the derived warp alignment timescale
(equation 20), while the dashed blue line shows the prediction from the same
approach, but using the alignment timescale from the differential equation
approach, given by equation (19). The agreement between the black and dashed
blue lines shows that the two approaches are (almost) equivalent.

##  Appendix C Additional parameter variations

Here we provide results on and discuss variations of different parameters
related to jet feedback and our setup. These simulations were all of our high-
mass galaxy cluster ($M_{\mathrm{200}}=10^{15}$ $\mathrm{M}_{\odot}$),
simulated at medium resolution ($m_{\mathrm{g}}=6.4\times 10^{6}$
$\mathrm{M}_{\odot}$).

###  C.1 Direction of jet launching: opening angle and inclination

In the top row of Fig. 12 we show the effects of varying the half-opening
angle of the jets (the fiducial value being $\theta_{\mathrm{j}}=10\degree)$.
We find that all relevant quantities are very similar, even with a ballistic
jet ($\theta_{\mathrm{j}}=0\degree$). This is somewhat surprising, given the
fact that jets with smaller opening angles are able to reach larger distances
(e.g. Kaiser & Best 2007, Huško & Lacey 2022). The only visible difference we
find is that the ballistic jet appears to feature less reorientation relative
to the initial direction (visible in the plot of the misalignment angle). In
addition, the spin of the SMBH takes on larger values, on average. In the
bottom row of Fig. 12 we show results of varying the angle between the angular
momentum vector of the halo and the initial SMBH spin vector of the SMBH (the
fiducial angle being $\theta_{0}=0\degree$). The results are again very
similar in terms of jet power, SFR and cold gas. There are no discernible
differences among these quantities with different misalignment angles. This is
likely indicative of the chaotic nature of these simulations.

Figure 12: Time dependence of the quenching/feedback process in the high-mass
galaxy cluster simulations ($M_{\mathrm{200}}=10^{15}$ $\mathrm{M}_{\odot}$)
with varying half-opening angles of the jet (top) and initial inclinations of
the SMBH spin vector, relative to the angular momentum of the halo (bottom).
The left-hand panels show the jet power, while the right-hand panels show the
star formation rate. Insets in the left-hand panels show the magnitude of SMBH
spin and the angle between the spin vector and $z-$axis. The insets in the
right-hand panels show the cold gas mass. The fiducial simulation is the one
with half-opening angle $\theta_{\mathrm{j}}=10\degree$ and inclination
$\theta_{0}=0\degree$ (purple and blue line, respectively, in the top and
bottom panels); other details of the fiducial setup are given in Table 1. The
dotted black lines represent the upper limit of the specific star formation
rate required to classify a galaxy as quenched.

###  C.2 Impact of ICM properties

In the top row of Fig. 13 we show results of varying the spin parameter of the
gaseous halo of our galaxy cluster. Our fiducial value is
$\lambda_{\mathrm{g}}=0.05$, and here we show cases with values that are equal
to half and double our fiducial, as well as no net rotation of the halo. The
differences between these cases are minor, especially in terms of jet power,
cold gas or SFR. The case with largest rotation ($\lambda_{\mathrm{g}}=0.1$)
features on average somewhat higher spins and is never significantly
misaligned from the $z$ axis. Overall, these results imply that the rotation
of the halo does not play a very significant role, and that the behaviour of
the cooling gas is chaotic. These conclusions should be treated carefully,
however, since these simulations were performed with a resolution of
$6.4\times 10^{6}$ $\mathrm{M}_{\odot}$, which means that the cold gas is
sampled with $100-1000$ particles at most.

Figure 13: Time dependence of the quenching/feedback process in the high-mass
galaxy cluster simulations ($M_{\mathrm{200}}=10^{15}$ $\mathrm{M}_{\odot}$)
with varying spin parameters (top) and metallicity profiles of the ICM
(bottom). The left-hand panels show the jet power, while the right-hand panels
show the star formation rate. Insets in the left-hand panels show the
magnitude of SMBH spin and the angle between the spin vector and $z$-axis. The
insets in the right-hand panels show the cold gas mass. The fiducial
simulation is the one with an ICM spin parameter of
$\lambda_{\mathrm{g}}=0.05$ and constant metallicity $Z=0.3Z_{\odot}$ (orange
and purple lines, respectively, in the top and bottom panels); other details
of the fiducial setup are given in Table 1. The dotted black lines represent
the upper limit of the specific star formation rate required to classify a
galaxy as quenched.

In the bottom row of Fig. 13 we show variations in metallicity profiles of
gas. We compare our fiducial case, with a constant metallicity of
$0.3Z_{\odot}$, against a case with a ’spread’ metallicity profile in accord
with observations, $Z(r)=Z_{\odot}/(1+r/20\hskip
0.85358pt\mathrm{kpc})^{0.26}$ (which falls from solar values to
$0.3Z_{\odot}$ at the virial radius), a ’concentrated’ profile
$Z(r)=Z_{\odot}/(1+r/20\hskip 0.85358pt\mathrm{kpc})^{0.5}$ (which falls to
$0.3Z_{\odot}$ by 100 kpc), and a similar one (’3xconcentrated’) that has a
central metallicity of $3Z_{\odot}$, but also falls to $0.3Z_{\odot}$ by $100$
kpc. The ’spread’ case features the highest SFR of all the simulations. The
two other cases do not show significant differences relative to the fiducial
case, despite the central metallicity being $\approx 3$ and $10$ times higher.
This is likely because the higher metallicities cause faster cooling only for
$T<$ a few $\times$ $10^{6}$ K. However, gas that has already cooled down to
such temperatures is already rapidly cooling, so the higher metallicities have
an effect only on how quickly this portion of cooling occurs. The cooling gas
spends the vast majority of its time at $T>10^{7}$ K, so metallicity plays
only a small role in these simulations.

###  C.3 Impact of other variations

In the top row of Fig. 14, we show the effects of reducing the gravitational
softening length from $1200$ pc, our fiducial choice, to $600$ and $300$ pc.
These simulations are very similar. The one with the lowest softening length
reaches the highest peak in SFR, but that is possibly incidental and due to
the chaotic nature of these massive galaxy cluster simulations. The spin
evolution in the three simulations is remarkably similar, despite the fact
that the evolution of the misalignment between the spin vector and $z-$axis is
not. Reducing the softening length does not appear to result in more coherent
rotating, cold gas discs near the SMBH.

Figure 14: Time dependence of the quenching/feedback process in high-mass
galaxy cluster simulations ($M_{\mathrm{200}}=10^{15}$ $\mathrm{M}_{\odot}$)
with varying gravitational softening lengths at fixed mass resolution (top)
and particle position seeding (bottom). The left-hand panels show the jet
power, while the right-hand panels show the star formation rate. Insets in the
left-hand panels show the magnitude of SMBH spin and the angle between the
spin vector and $z$-axis. The insets in the right-hand panels show the cold
gas mass. The fiducial simulation is the one with a gravitational softening
length of of $\epsilon_{\mathrm{g}}=1.2$ kpc and random particle seeding
(purple line in each panel); other details of the fiducial setup are given in
Table 1. The dotted black lines represent the upper limit of the specific star
formation rate required to classify a galaxy as quenched.

In the bottom row of Fig. 14 we compare two simulations with different initial
conditions: random seeding (our fiducial choice) versus a uniform cubic
lattice of particles, that is then stretched to produce the desired density
profile. The differences are clearly visible. With uniform initial conditions,
the gas cools more rapidly and there are generally more frequent jet/star
formation episodes. This is likely because cooling caused by thermal
instabilities from random seeding is not present, so the eventual cooling
flows are stronger and harder to suppress (we find that overall more jet
energy is injected in this simulation). Despite the lack of initial
randomness, even the uniform simulation results in spindown and misalignment
of the spin vector (with turbulence possibly being caused by jet feedback
itself).


#  ${\mathcal{K}}$-Lorentzian Polynomials

Grigoriy Blekherman and Papri Dey  School of Mathematics, Georgia Institute of
Technology, Atlanta, GA 30332

###### Abstract.

Lorentzian polynomials are a fascinating class of real polynomials with many
applications. Their definition is specific to the nonnegative orthant.
Following recent work, we examine Lorentzian polynomials on proper convex
cones. For a self-dual cone ${\mathcal{K}}$ we find a connection between
${\mathcal{K}}$-Lorentzian polynomials and ${\mathcal{K}}$-positive linear
maps, which were studied in the context of the generalized Perron-Frobenius
theorem. We find that as the cone ${\mathcal{K}}$ varies, even the set of
quadratic ${\mathcal{K}}$-Lorentzian polynomials can be difficult to
understand algorithmically. We also show that, just as in the case of the
nonnegative orthant, ${\mathcal{K}}$-Lorentzian and ${\mathcal{K}}$-completely
log-concave polynomials coincide.

###### Key words and phrases:

Lorentzian polynomials, convex cone, log-concavity

###### 2020 Mathematics Subject Classification:

14P99, 52A20

##  1\. Introduction

Lorentzian polynomials were introduced by Brändén and Huh in [BH20] and
simultaneously by Anari, Oveis Gharan and Vinzant [AGV21] (under the name of
completely log-concave polynomials) as a generalization of hyperbolic
polynomials that retain their log-concavity properties. They have already seen
a multitude of applications [ALOGV24a, ALOGV24b, AD20]. Lorentzian polynomials
are specific to the nonnegative orthant $\mathbb{R}^{n}_{\geq 0}$. Lorentzian
polynomials with respect to a full-dimensional convex cone
${\mathcal{K}}\subset{\mathbb{R}}^{n}$ were introduced by Brändén and Leake in
[BL23] and by the second author.

We consider what interesting properties of Lorentzian polynomials are
maintained when we switch from the nonnegtave orthant to more general cones.
The cone ${\mathbb{R}}^{n}_{\geq 0}$ is self-dual, and for any self-dual cone
${\mathcal{K}}$ there is an immediate connection between
${\mathcal{K}}$-Lorentzian polynomials and well-studied concept of
${\mathcal{K}}$-nonnegative linear maps [Bv20]. This allows us to also
consider properties of ${\mathcal{K}}$-irreducibility and
${\mathcal{K}}$-positivity, which have been studied in the context of
generalizing the Perron-Frobenius theorem [Gan98, Van68].

We take a detailed look at ${\mathcal{K}}$-Lorentzian quadratic forms for
various self-dual cones ${\mathcal{K}}$. Quadratic Lorentzian polynomials are
simple to understand, and so are quadratic forms Lorentzian on the second
order cone $\mathcal{L}_{n}$ which can be characterized via a simple
semidefinite program. However, quadratic forms Lorenztian on the cone of
positive semidefinite matrices $\mathcal{S}^{n}_{+}$ are more difficult. We
show that a natural linear projection of $\mathcal{S}^{n}_{+}$-Lorentzian
quadratic forms is the convex cone of degree $4$ globally nonnegative
polynomials. From this we see that testing whether a quadratic form is
$\mathcal{S}^{n}_{+}$-Lorentzian is NP-hard.

In analogy with results of [BH20] and [ALOGV24b] we show that
${\mathcal{K}}$-Lorentzian polynomials are equivalently completely log-concave
polynomials (CLC) on ${\mathcal{K}}$. We also find some interesting properties
of Hessians of ${\mathcal{K}}$-Lorentzian polynomials related to
${\mathcal{K}}$-nonnegativity and ${\mathcal{K}}$-irreducibility. We provide a
different proof of a result of [BL23] that testing whether a polynomial is
${\mathcal{K}}$-Lorentzian can be reduced to taking directional derivatives
with respect to extreme rays of ${\mathcal{K}}$ and one interior point of
${\mathcal{K}}$, and examine the interior of the set of
${\mathcal{K}}$-Lorentzian polynomials. The rest of the paper is structured as
follows: in Section 2 we set notation and main definitions. Section 3 is
devoted to a detailed examination of quadratic ${\mathcal{K}}$-Lorentzian
polynomials; we also make a first connection to ${\mathcal{K}}$-nonnegative
and ${\mathcal{K}}$-irreducible matrices. In Section 4 we examine the interior
of the set of ${\mathcal{K}}$-Lorentzian polynomials, show equivalence between
${\mathcal{K}}$-Lorentzian and ${\mathcal{K}}$-completely log-concave
polynomials and provide a sufficient condition for a sum of two
${\mathcal{K}}$-Lorentzian polynomials to be ${\mathcal{K}}$-Lorentzian. In
Section 5 we expand on the connection to ${\mathcal{K}}$-irreducibility and
${\mathcal{K}}$-nonnegativity and show a simplified characterization of
${\mathcal{K}}$-Lorentzian polynomials.

##  2\. Definitions and Notation

A nonempty convex set ${\mathcal{K}}\subseteq{\mathbb{R}}^{n}$ is said to be a
cone if $c{\mathcal{K}}\subseteq{\mathcal{K}}$ for all $c\geq 0$. A cone
${\mathcal{K}}$ is called proper if it is closed (in the Euclidean topology on
${\mathbb{R}}^{n}$), pointed (i.e.,
${\mathcal{K}}\cap(-{\mathcal{K}})=\\{0\\}$), and solid (i.e., the topological
interior of ${\mathcal{K}}$, denoted as $\operatorname{int}{\mathcal{K}}$, is
nonempty).

The vector space of real homogeneous polynomials (forms) in $n$ variables of
degree $d$ is denoted by ${\mathbb{R}}[x]^{d}_{n}$. A form
$f\in{\mathbb{R}}[x]_{n}^{d}$ is said to be _log-concave_ at a point
$a\in{\mathbb{R}}^{n}$ if $f(a)>0$ and $\log f$ is a convex function at $a$,
i.e. the Hessian of $\log f$ is negative semidefinite at $a$. A form $f$ is
_strictly log-concave_ at $a$ if the Hessian of $\log f$ at $a$ is negative
definite. A form $f\in{\mathbb{R}}[x]_{n}^{d}$ is log-concave on a proper cone
${\mathcal{K}}\subset{\mathbb{R}}^{n}$ if $f$ is log-concave at every point of
the interior of ${\mathcal{K}}$. By convention, the zero polynomial is log-
concave (but not strictly log-concave) at all points of ${\mathbb{R}}^{n}$.

We will denote quadratic forms by a lower case letter, and the matrix of the
quadratic form by the corresponding upper case letter, i.e. if $q$ is a
quadratic form, then its matrix is $Q$ and $q(x)=x^{t}Qx$.

For a point $a\in{\mathbb{R}}^{n}$ and $f\in{\mathbb{R}}[x]_{n}^{d}$ we use
$D_{a}f$ to denote the directional derivative of $f$ in direction $a$:
$D_{a}f=\sum_{i=1}^{n}a_{i}\frac{\partial f}{\partial x_{i}}$. We now define
${\mathcal{K}}$-completely log-concave and ${\mathcal{K}}$-Lorentzian forms.

######  Definition 2.1.

A form $f\in{\mathbb{R}}[x]_{n}^{d}$ is called a ${\mathcal{K}}$-completely
log-concave (${\mathcal{K}}$-CLC) on a proper convex cone ${\mathcal{K}}$ if
for any choice of $a_{1},\dots,a_{m}\in{\mathcal{K}}$, with $m\leq d$, we have
that $D_{a_{1}}\dots D_{a_{m}}f$ is log-concave on
$\operatorname{int}{\mathcal{K}}$. A form $f\in{\mathbb{R}}[x]_{n}^{d}$ is
strictly ${\mathcal{K}}$-CLC if for any choice of
$a_{1},\dots,a_{m}\in{\mathcal{K}}$, with $m\leq d$, $D_{a_{1}}\dots
D_{a_{m}}f$ is strictly log-concave on all points of ${\mathcal{K}}$. We use
$\operatorname{\mathrm{CLC}}^{d}({\mathcal{K}})$ to denote the set of
${\mathcal{K}}$-CLC forms in ${\mathbb{R}}[x]_{n}^{d}$.

######  Definition 2.2.

Let ${\mathcal{K}}$ be a proper convex cone. A form
$f\in{\mathbb{R}}[x]_{n}^{d}$ of degree $d\geq 2$ is said to be
${\mathcal{K}}$-Lorentzian if for any
$a_{1},\dots,a_{d-2}\in\operatorname{int}{\mathcal{K}}$ the quadratic form
$q=D_{a_{1}}\dots D_{a_{d-2}}f$ satisfies the following conditions:

  1. (1)

The matrix $Q$ of $q$ has exactly one positive eigenvalue.

  2. (2)

For any $x,y\in\operatorname{int}{\mathcal{K}}$ we have $y^{t}Qx=\langle
y,Qx\rangle>0$.

For degree $d\leq 1$ a form is ${\mathcal{K}}$-Lorentzian if it is nonnegative
on ${\mathcal{K}}$. We use $\operatorname{\mathrm{SL}}^{d}({\mathcal{K}})$ to
denote the set of ${\mathcal{K}}$-Lorentzian forms in
${\mathbb{R}}[x]_{n}^{d}$.

######  Remark 2.3.

Definition 2.2 is equivalent to the definition in [BL23], although we state it
in a slightly different way. For a quadratic form $q$ with matrix $Q$ and for
all $a,b\in{\mathbb{R}}^{n}$ we have $D_{b}D_{a}q=2b^{t}Qa$. Therefore, we can
also express condition (2) as $D_{b}D_{a}q>0$ for all
$a,b\in\operatorname{int}{\mathcal{K}}$. Combining with the fact that we have
the freedom to choose directions of derivatives
$a_{1},\dots,a_{d-2}\in\operatorname{int}{\mathcal{K}}$ we can express
condition (2) as $D_{a_{1}}\dots D_{a_{d}}f>0$ for all
$a_{1},\dots,a_{d}\in\operatorname{int}{\mathcal{K}}$, as was done in [BL23].

We can also write conditions (1) and (2) above without relying on writing the
matrix of $q$ in coordinates. Condition (1) can be stated via the variational
definition of eigenvalues: there exists a point $x\in{\mathbb{R}}^{n}$ such
that $q(x)>0$, and for any two dimensional subspace $V$ of $R^{n}$, there
exists $v\in V$, $v\neq 0$ such that $q(v)\leq 0$. Condition (2) can also be
stated in terms of the naturally associated bilinear form $B(x,y)$ of $q(x)$:
there exists a unique bilinear form $B(x,y)$ such that $q(x)=B(x,x)$ for all
$x\in{\mathbb{R}}^{n}$. We can equivalently say that $B(x,y)>0$ for all
$x,y\in\operatorname{int}{\mathcal{K}}$.

##  3\. Quadratic ${\mathcal{K}}$-Lorentzian polynomials

In this section we analyze properties of quadratic ${\mathcal{K}}$-Lorentzian
polynomials. Although these observations are mostly elementary, they point to
interesting connections with other well-studied notions. We first describe
several equivalent formulations for nonnegativity condition (2) in Definition
2.2, which may be easier to verify.

######  Lemma 3.1.

A quadratic form $q(x)\in{\mathbb{R}}[x]_{n}^{2}$ is
${\mathcal{K}}$-Lorentzian on a proper convex cone ${\mathcal{K}}$ if and only
if the matrix $Q$ of $q$ has exactly one positive eigenvalue and one of the
following conditions holds:

  1. (a)

for all $x\in{\mathcal{K}}$ we have $Qx\in{\mathcal{K}}^{*}$, i.e.
$Q({\mathcal{K}})\subseteq{{\mathcal{K}}^{*}}$.

  2. (b)

$y^{t}Qx\geq 0$ for all $x,y$ spanning extreme rays of ${\mathcal{K}}$.

  3. (c)

$x^{t}Qx>0$ for all $x\in\operatorname{int}{\mathcal{K}}$.

######  Remark 3.2.

Note that conditions (b) and (c) of Lemma 3.1 are relaxations of Condition (2)
from LABEL:{def:pls}.

###### Proof.

(a) By the Biduality Theorem [Bar02, Chapter-IV] $\langle{y,Qx}\rangle\geq 0$
for all $y\in{\mathcal{K}}$ if and only if $Qx\in{\mathcal{K}}^{*}$. If
$Q({\mathcal{K}})=\\{0\\}$, then $Q$ is the zero matrix and $q$ is
${\mathcal{K}}$-Lorentzian. If $Q$ is not the zero matrix and $Qx=0$ for
$x\in\operatorname{int}{\mathcal{K}}$, then $Q({\mathcal{K}})$ contains a
linear subspace. Since $Q({\mathcal{K}})\subseteq{\mathcal{K}}^{*}$ we see
that ${\mathcal{K}}$ is not full-dimensional, which is a contradiction.
Therefore $Qx\neq 0$ for any $x\in\operatorname{int}{\mathcal{K}}$. Then for
any $y\in\operatorname{int}K$ we have $\langle y,Qx\rangle>0$.
(b) Since ${\mathcal{K}}$ is a proper convex cone, by the conical Minkowksi
Theorem [Bar02, Chapter II], it is the conical hull of its extreme rays.
Therefore $Q({\mathcal{K}})\subseteq{\mathcal{K}}^{*}$ if and only if for all
$x\in{\mathcal{K}}$ spanning extreme rays of ${\mathcal{K}}$ we have
$Qx\subset{\mathcal{K}}^{*}$. Additionally, we have $Qx\in{\mathcal{K}}^{*}$
if and only if for all $y$ spanning extreme rays of ${\mathcal{K}}$ we have
$y^{t}Qx\geq 0$.
(c) Let $v=(v_{1},\dots,v_{n})\in\operatorname{int}{\mathcal{K}}$. Let $C$ be
the connected component of the set
$\\{x\in{\mathbb{R}}^{n}\,\,\mid\,\,q(x)>0\\}$ which contains $v$. Then $C$ is
an open convex cone [G5̇9], [G9̈7] (which is the hyperbolicity cone of $q$),
and its closure $\bar{C}$ is a closed, convex cone. Note that we have
${\mathcal{K}}\subset\bar{C}$, and we need to show that
$Q({\mathcal{K}})\subseteq{\mathcal{K}}^{*}$. Since duality reverses
inclusion, we have $\bar{C}^{*}\subseteq{\mathcal{K}}^{*}$. Therefore it
suffices to show that $Q(\bar{C})\subseteq\bar{C}^{*}$.

Since $Q$ has exactly one positive eigenvalue, then up to an orthogonal change
of variables, we can write
$Q(x)=\lambda_{1}x_{1}^{2}-\lambda_{2}x_{2}^{2}-\dots-\lambda_{k}x_{k}^{2}$
with $k\leq n$ and $\lambda_{i}>0$. Furthermore we may assume $v_{1}>0$. Let
$q^{\prime}$ be the quadratic form
$q^{\prime}(x)=\frac{1}{\lambda_{1}}x_{1}^{2}-\frac{1}{\lambda_{2}}x_{2}^{2}-\dots-\frac{1}{\lambda_{k}}x_{k}^{2}$.
We have

$\displaystyle\bar{C}=\\{(x_{1},\dots,x_{n})\in{\mathbb{R}}^{n}\,\,\mid\,\,x_{1}\geq
0,\,q(x)\geq 0\\},$
$\displaystyle\bar{C}^{*}=\\{(x_{1},\dots,x_{n})\in{\mathbb{R}}^{n}\,\,\mid\,\,x_{1}\geq
0,\,q^{\prime}(x)\geq 0,\,x_{k+1}=\dots=x_{n}=0\\}.$

Take $x=(x_{1},\dots,x_{n})\in\bar{C}$. Let
$y=Qx=(\lambda_{1}x_{1},-\lambda_{2}x_{2},\dots,-\lambda_{k}x_{k},0,\dots,0)$.
Observe that $y_{1}\geq 0$, $y_{k+1}=\dots=y_{n}=0$ and
$q^{\prime}(y)=q(x)\geq 0$. Therefore $y\in\bar{C}^{*}$ and the Lemma follows.

∎

Next we show that nonnegativity condition (2) from Definition 2.2 can be even
further relaxed to just testing nonnegativity of $q$ on the extreme rays of
${\mathcal{K}}$ in case the matrix $Q$ is nonsingular, and the set of extreme
rays of ${\mathcal{K}}$ is path-connected.

######  Lemma 3.3.

Let $q$ be a a quadratic form on ${\mathbb{R}}^{n}$ such that its matrix $Q$
is nonsingular and has exactly one positive eigenvalue. Let ${\mathcal{K}}$ be
a proper convex cone in ${\mathbb{R}}^{n}$ such that the set of vectors
spanning an extremal ray of ${\mathcal{K}}$ is path-connected. Then
$q\in\operatorname{\mathrm{SL}}^{2}({\mathcal{K}})$ if and only if $q(v)\geq
0$ for all $v$ spanning an extremal ray of $K$.

######  Remark 3.4.

The reduction to extreme rays is not possible for the nonnegative orthant
${\mathbb{R}}^{n}_{\geq 0}$ (the standard setting for Lorentzian polynomials),
or for any other polyhedral cone ${\mathcal{K}}$. In these cases, the set of
vectors spanning extreme rays of ${\mathcal{K}}$ is not connected.

###### Proof.

Up to a change of coordinates we have $q(x)=x_{1}^{2}-x_{2}^{2}-\dots-
x_{n}^{2}$. Let $v,w$ be two points spanning distinct extreme rays of
${\mathcal{K}}$. We see that the first coordinates $v_{1}$, $w_{1}$ must have
the same sign, since there is a path connecting $v$ and $w$ which does not
pass through the origin, such that for any point $t$ on the path we have
$q(t)\geq 0$. Without loss of generality we may assume that for any $v$
spanning an extremal ray of ${\mathcal{K}}$ we have $v_{1}>0$. It follows that
all extreme rays of ${\mathcal{K}}$ are contained in the proper convex cone
$\bar{C}=\\{(x_{1},\dots,x_{n})\in{\mathbb{R}}^{n}\,\,\mid\,\,x_{1}\geq
0,\,q(x)\geq 0\\}$. Therefore, the interior of ${\mathcal{K}}$ is contained in
the interior of $\bar{C}$ and $q$ is strictly positive on the interior of
$\bar{C}$. The Lemma now follows by Lemma 3.1. ∎

######  Remark 3.5.

The hypothesis that $Q$ is nonsingular cannot be eliminated. Let $\ell$ be any
linear form such that $\ell(v)=0$ for some
$v\in\operatorname{int}{\mathcal{K}}$, and let $q=\ell(v)^{2}$. Then $Q$ has
exactly one positive eigenvalue, and $q$ is nonnegative on all the extreme
rays of ${\mathcal{K}}$, but $q$ is not ${\mathcal{K}}$-Lorentzian.

Let ${\mathcal{K}}$ be a self-dual cone, i.e.
${\mathcal{K}}={\mathcal{K}}^{*}$. We can use Condition (a) of Lemma 3.1 to
make a connection between ${\mathcal{K}}$-Lorentzian quadratic forms and
${\mathcal{K}}$-nonnegative matrices (or equivalently linear maps).

######  Definition 3.6.

Let ${\mathcal{K}}$ be a proper convex cone in ${\mathbb{R}}^{n}$. A $n\times
n$ matrix $A$ is ${\mathcal{K}}$-nonnegative if $Ax\in{\mathcal{K}}$ for any
$x\in{\mathcal{K}}$, i.e. $A({\mathcal{K}})\subseteq{\mathcal{K}}$. A matrix
$A$ is ${\mathcal{K}}$-positive if $Ax\in\operatorname{int}{\mathcal{K}}$ for
any nonzero $x\in{\mathcal{K}}$. A ${\mathcal{K}}$-nonnegative matrix $A$ is
called ${\mathcal{K}}$-irreducible if $A$ leaves no (proper) face of
${\mathcal{K}}$ invariant. A ${\mathcal{K}}$-nonnegative matrix which is not
${\mathcal{K}}$-irreducible is called ${\mathcal{K}}$-reducible, cf. [Gan98],
[Van68].

Here we record some equivalent characterizations to
${\mathcal{K}}$-irreducibility which will be used in the sequel [Van68].

######  Theorem 3.7.

Let $A$ be a ${\mathcal{K}}$-nonnegative matrix for a solid (with nonempty
interior) cone ${\mathcal{K}}$. The following statements are equivalent:

  1. (1)

$A$ is ${\mathcal{K}}$-irreducible

  2. (2)

no eigenvector of $A$ lies on the boundary $\partial{\mathcal{K}}$ of
${\mathcal{K}}$.

  3. (3)

$A$ has exactly one eigenvector in ${\mathcal{K}}$, and this eigenvector is in
$\operatorname{int}{\mathcal{K}}$.

From Lemma 3.1 (a) we immediately see the following:

######  Corollary 3.8.

Let ${\mathcal{K}}$ be a self-dual cone. Then
$q\in\operatorname{\mathrm{SL}}^{2}({\mathcal{K}})$ if and only if the matrix
$Q$ of $q$ has exactly one positive eigenvalue and $Q$ is
${\mathcal{K}}$-nonnegative.

We now show that if a matrix $Q$ of
$q\in\operatorname{\mathrm{SL}}^{2}({\mathcal{K}})$ is nonsingular and the
cone ${\mathcal{K}}$ is self-dual, then $Q$ must be
${\mathcal{K}}$-irreducible.

######  Lemma 3.9.

Let ${\mathcal{K}}\subset{\mathbb{R}}^{n}$ be a self-dual cone and
$q\in\operatorname{\mathrm{SL}}^{2}({\mathcal{K}})$ with matrix $Q$. If $Q$ is
nonsingular, then $Q$ is ${\mathcal{K}}$-irreducible.

###### Proof.

It suffices to show that a symmetric, nonsingular ${\mathcal{K}}$-nonnegative
matrix $Q$ with exactly one positive eigenvalue must be
${\mathcal{K}}$-irreducible. Let $v$ be the eigenvector associated with the
positive eigenvalue $\alpha$ of $Q$. Suppose that neither $v$ nor $-v$ lie in
the interior of ${\mathcal{K}}$. Since ${\mathcal{K}}$ is self-dual, there
exists a nonzero vector $w\in{\mathcal{K}}$ such that $\langle v,w\rangle=0$.
Since $Q$ is nonsingular, we see that the quadratic form $q$ is negative
definite on the orthogonal complement of $v$, and in particular, we have
$q(w)<0$. This is a contradiction, since $q$ is nonnegative on
${\mathcal{K}}$. The Lemma now follows by Theorem 3.7. ∎

Combining Corollary 3.8 and Lemma 3.9 we immediately obtain the following:

######  Corollary 3.10.

Let ${\mathcal{K}}\subset{\mathbb{R}}^{n}$ be a self-dual cone. Then
$q\in\operatorname{\mathrm{SL}}^{2}({\mathcal{K}})$ if and only if the matrix
$Q$ of $q$ has exactly one positive eigenvalue and $Q$ is either nonsingular
and ${\mathcal{K}}$-irreducible, or singular and ${\mathcal{K}}$-nonnegative.

###  3.1. Quadratic Lorentzian Polynomials on ${\mathbb{R}}^{n}_{\geq 0}$ and
the Second order cone.

Quadratic forms Lorentzian on the nonnegative orthant ${\mathbb{R}}^{n}_{\geq
0}$ (the usual Lorenztian polynomials) are simply quadratic forms with
nonnegative coefficients and exactly one positive eigenvalue (see e.g. Lemma
3.1 (b)). Therefore it is easy to test whether a quadratic form is
${\mathbb{R}}^{n}_{\geq 0}$-Lorentzian. It was recently shown in [Chi24] that
while testing whether a polynomial is Lorentzian is easy in any degree,
testing whether a polynomial is stable (hyperbolic on the nonnegative orthant)
is a diffciult computational problems starting in degree $3$.

The second order cone $\mathcal{L}_{n}\subset{\mathbb{R}}^{n}$ is defined by

$\mathcal{L}_{n}=\\{(x_{1},\dots,x_{n})\in{\mathbb{R}}^{n}\,\,\mid\,\,x_{1}^{2}\geq
x_{2}^{2}+\dots+x_{n}^{2}\\}.$

Let $B$ be the diagonal matrix $n\times n$ matrix with $1$ in the
$(1,1)$-entry and $-1$ in all other diagonal entries. It is well-known that by
the S-Lemma [PT07], [LS75] a quadratic form $q$ with matrix $Q$ is nonnegative
on $\mathcal{L}_{n}$ if and only if there exists a nonnegative
$\lambda\in{\mathbb{R}}$ such that $Q-\lambda B$ is positive semidefinite.
Note that $\lambda>0$ would imply that $q$ is strictly positive on the
interior of $\mathcal{L}_{n}$, and therefore if $q$ has exactly one positive
eigenvalue, then $q$ is $\mathcal{L}_{n}$-Lorentzian (Lemma 3.1 (c)). If
$\lambda=0$, then $Q$ is positive semidefinite, and if $Q$ has exactly one
positive eigenvalue, then $q=\langle v,x\rangle^{2}$ for some
$v\in{\mathbb{R}}^{n}$. In this case $q$ is $\mathcal{L}_{n}$-Lorentzian if
and only if $v\in\mathcal{L}_{n}$, since the second order cone is self-dual.
Therefore we can test whether a quadratic form is $\mathcal{L}_{n}$-Lorentzian
by semidefinite programming and establishing the signature of $Q$.

However, as we will see below, there are self-dual cones ${\mathcal{K}}$ for
which the set of ${\mathcal{K}}$-Lorenztian quadratic forms is complicated. We
now turn to the cone of positive semidefinite matrices $\mathcal{S}^{n}_{+}$.

###  3.2. Quadratic Lorentzian Polynomials on $\mathcal{S}^{n}_{+}$

We consider quadratic forms Lorentzian on the cone $\mathcal{S}^{n}_{+}$ of
$n\times n$ positive semidefinite symmetric matrices. Let $\mathcal{S}^{n}$
denote the vector space of real symmetric $n\times n$ matrices equipped with
the standard inner product $\langle A,B\rangle=\operatorname{trace}AB$. It is
well-known that the cone $\mathcal{S}^{n}_{+}$ is self-dual with the respect
to the standard inner product [Bar02, Chapter II.12]. Linear maps that are
$S^{n}_{+}$-nonnegative are known as _positive linear maps_ , and they are
also related to nonnegative biquadratic forms on
$\mathbb{R}^{n}\times\mathbb{R}^{n}$. We describe this connection now.

Let $V$ be the vector space of linear maps from $\mathcal{S}^{n}$ to itself.
Note that here we are not restricting the matrices of linear transformations
to be symmetric, and we have $\dim V=\binom{n}{2}^{2}$. A linear map
$L:\mathcal{S}^{n}\rightarrow\mathcal{S}^{n}$ is called _a positive linear
map_ if $L(\mathcal{S}^{n}_{+})\subseteq\mathcal{S}^{n}_{+}$. It follows from
Corollary 3.8 that quadratic $\mathcal{S}^{n}_{+}$-Lorentzian polynomials
correspond to positive linear maps whose matrix is symmetric and has exactly
one positive eigenvalue.

Let $W$ be the vector of real biquadaratic forms on
$\mathbb{R}^{n}\times\mathbb{R}^{n}$, i.e. homogeneous polynomials in
variables $(x_{1},\dots,x_{n})$, $(y_{1},\dots,y_{n})$ which are quadratic in
$x$ and $y$ variables. Note that $\dim W=\binom{n}{2}^{2}$. We can define a
linear map $\Psi$ from $V$ to $W$.

$\Psi:V\rightarrow W,\,\,\,\text{via}\,\,\,\Psi(L)=\langle
yy^{t},L(xx^{t})\rangle.$

It is well-known that $\Psi$ is a linear isomorphism and a map
$L:\mathcal{S}^{n}\rightarrow\mathcal{S}^{n}$ is positive if and only if the
associated biquadratic form $\Psi(L)$ is nonnegative on
$\mathbb{R}^{n}\times\mathbb{R}^{n}$ [Bv20].

Let $V^{\prime}$ be the vector space of quadratic forms on $\mathcal{S}^{n}$,
which we can equivalently see as a subspace of $V$ consisting of linear maps
$\mathcal{S}^{n}\rightarrow\mathcal{S}^{n}$ with a symmetric matrix. It
follows that $\Psi(V^{\prime})$ is the vector space $W^{\prime}$ of
biquadratic forms which are symmetric under swapping $x$ and $y$. We see from
the above discussion that the image of
$\operatorname{\mathrm{SL}}^{2}(\mathcal{S}^{n}_{+})$ under $\Psi$ is a non-
convex subset of the cone of nonnegative biquadratic forms in $W^{\prime}$. It
is not clear to us if the signature condition of exactly one positive
eigenvalue has a natural translation in terms of biquadratic forms.

However, we can use the criterion of Lemma 3.3 to provide a simpler criterion
for nonsingular quadratic forms to be $\mathcal{S}^{n}_{+}$-Lorentzian , which
will give us a different linear map, where we will have full understanding of
the image of $\operatorname{\mathrm{SL}}^{2}(\mathcal{S}^{n}_{+})$

######  Corollary 3.11.

Let $q$ be a quadratic form on $\mathcal{S}^{n}$ whose matrix $Q$ is
nonsingular and has exactly one positive eigenvalue. Then $q$ is
$\mathcal{S}^{n}_{+}$-Lorentzian if and only if the associated quartic
polynomial $q(xx^{t})$ is nonnegative for all $x\in\mathbb{R}^{n}$.

###### Proof.

The extreme rays of the cone $\mathcal{S}^{n}_{+}$ are well-known to be
spanned by rank $1$ matrices of the form $xx^{t}$ for $x\in{\mathbb{R}}^{n}$
[Bar02, Chapter II.12]. Now we apply Lemma 3.3. ∎

Motivated by the above result we define the linear map
$\Phi:V^{\prime}\rightarrow\mathbb{R}[x]_{n}^{4}$ given by
$\Phi(q)=q(xx^{t})$. We can define a natural linear projection from the vector
space $W$ of biquadratic forms on ${\mathbb{R}}^{n}\times{\mathbb{R}}^{n}$ to
the vectors space of quartic polynomials ${\mathbb{R}}[x]_{n}^{4}$ by sending
a biquadratic form $T(x,y)$ to the quartic polynomial $T(x,x)$. We can see
that

$\Phi=\pi\circ\Psi,$

i.e. the quartic polynomial $q(xx^{t})$ can be obtained by plugging in $y=x$
in the associated biquadratic form $\Psi(q)$. Interestingly, the image of
$\operatorname{\mathrm{SL}}^{2}(\mathcal{S}_{+}^{n})$ under $\Phi$ is the
entire cone $P_{n,4}$ of nonnegative quartics on ${\mathbb{R}}^{n}$, and in
particular it is convex.

######  Theorem 3.12.

We have

$\Phi(\operatorname{\mathrm{SL}}^{2}(\mathcal{S}^{n}_{+}))=P_{n,4},$

and moreover the same holds if we replace
$\operatorname{\mathrm{SL}}^{2}(\mathcal{S}^{n}_{+})$ with nondegenerate
quadratic forms in $\operatorname{\mathrm{SL}}^{2}(\mathcal{S}^{n}_{+})$.

###### Proof.

It follows from the above discussion that the image of
$\operatorname{\mathrm{SL}}^{2}(\mathcal{S}^{n}_{+})$ under $\Phi$ lies inside
$P_{n,4}$.

Let $r$ be the following quadratic form on $\mathcal{S}^{n}$:

$r(X)=\sum_{1\leq i<j\leq n}x_{ii}x_{jj}-x^{2}_{ij},$

i.e. $r$ is the sum of all $2\times 2$ principal minors of a matrix
$X\in\mathcal{S}^{n}$. It follows that $\Phi(r)=r(xx^{t})=0$. If we
coordinatize $\mathcal{S}^{n}$ by first listing diagonal elements (in any
order) and then off-diagonal elements (in any order), then the matrix $R$ of
$r$ becomes:

$R=\begin{pmatrix}\begin{matrix}0&1/2&\dots&1/2&\\\ 1/2&0&\dots&1/2\\\
\vdots&\vdots&\vdots&\vdots\\\
1/2&1/2&\dots&0\end{matrix}&\makebox(0.0,0.0)[]{\text{\huge 0}}\\\
\makebox(0.0,0.0)[]{\text{\huge 0}}&\begin{matrix}-1&0&\dots&0\\\
0&-1&\dots&0\\\ \vdots&\vdots&\vdots&\vdots\\\
0&0&\dots&-1\end{matrix}\end{pmatrix}.$

We see that the matrix $R$ is nonsingular with exactly one positive
eigenvalue. Since the map $\Phi$ is onto ${\mathbb{R}}[x]_{n}^{4}$, given a
polynomial $p\in P_{n,4}$, there exists a quadratic form $a\in V^{\prime}$,
with matirx $A$, such that $\Phi(a)=p$. Now let $A^{\prime}=T+tR$ and let
$a^{\prime}$ be the associated quadratic form. For a sufficiently large $t$ we
see that $A^{\prime}$ is nonsingular with exactly one positive eigenvalue, and
therefore by Corollary 3.11 we have that
$a^{\prime}\in\operatorname{\mathrm{SL}}^{2}(\mathcal{S}^{n}_{+})$ and
$\Phi(a^{\prime})=\Phi(a)=p$. ∎

It quickly follows from the above result that testing whether a quadratic form
is $\mathcal{S}^{n}_{+}$-Lorentzian is NP-hard.

######  Corollary 3.13.

It is NP-hard to test whether a quadratic form is
$\mathcal{S}^{n}_{+}$-Lorentzian.

###### Proof.

Note that the projection $\Phi$ sending a quadratic form $q$ to $q(xx^{t})$
simply acts by substituting $x_{ij}=x_{i}x_{j}$ into $q$. Under this map,
several quadratic monomials $x_{ij}x_{k\ell}$ may map to the same degree 4
monomial, since for example, $x_{1}^{2}\cdot x_{2}^{2}=(x_{1}x_{2})^{2}$.
Therefore, in coordinates, $\Phi$ simply adds several coefficients of
monomials of $q$ to produce a coefficent of the quartic polynomial $\Phi(q)$,
and these sums are taken in disjoint groups.

Given a polynomial $p\in{\mathbb{R}}[x]_{n}^{4}$ we can define a canonical
preimage $m\in\Phi^{-1}(p)$ as follows. Let $x^{a}$ be a degree $4$ monomial
in $p$ with coefficient $c_{a}$. Define $d_{a}$ to be the number of possible
quadrartic monomial $x_{ij}x_{k\ell}$ that produce $x^{a}$ under the
substitution $x_{ij}=x_{i}x_{j}$. Then coefficient of $x_{ij}x_{k\ell}$ in $m$
is simply $\frac{c_{a}}{d_{a}}$ where $a=\Phi(x_{ij}x_{k\ell})$. By
construction we have $\Phi(m)=p$.

Then, as in proof of Theorem 3.12 we consider $M^{\prime}=M+tR$. It is not
hard to estimate $t$ such that $M^{\prime}$ has exactly one positive
eigenvalue. Therefore, if we can test in polynomial time whether quadratic
form $m^{\prime}$ associated to $M^{\prime}$ is
$\mathcal{S}^{n}_{+}$-Lorentzian, then we can test whether $p\in P_{n,4}$ by
Lemma 3.3. However, it is known that testing nonnegativity of degree 4
polynomials is NP-Hard [BCSS98, Part 1]. ∎

##  4\. Completely ${\mathcal{K}}$-log-concave and ${\mathcal{K}}$-Lorentzian
polynomials

We now investigate properties of the set
$\operatorname{\mathrm{SL}}^{d}({\mathcal{K}})$ of ${\mathcal{K}}$-Lorentzian
polynomials of degree $d$. We consider its interior, and we also show
equivalence between ${\mathcal{K}}$-Lorentzian and ${\mathcal{K}}$-CLC
polynomials.

###  4.1. Log-concavity over proper convex cones

We show several equivalent characterizations of forms log-concave on a proper
convex cone $\mathcal{K}\subset\mathbb{R}^{n}$. We employ the same reasoning
as outlined in [AGV21]. Additionally, we utilize the following version of
Cauchy’s interlacing Theorem.

######  Lemma 4.1 (Cauchy’s interlacing Theorem I).

[HJ13, Corollary $4.3.9$] Let $A$ be a symmetric matrix and vector
$v\in{\mathbb{R}}^{n\times n}$. Then, the eigenvalues of $A-vv^{t}$ interlace
the eigenvalues of $A$, i.e., if $\alpha_{1},\dots,\alpha_{n}$ are eigenvalues
of $A$ and $\beta_{1},\dots,\beta_{n}$ are eigenvalues of $A-vv^{T}$, then

$\beta_{1}\leq\alpha_{1}\leq\dots\beta_{n}\leq\alpha_{n}$

Therefore, we have the following result.

######  Lemma 4.2.

[AGV21, Lemma 2.4] Let $Q\in{\mathbb{R}}^{n\times n}$ be a symmetric matrix
and let $P\in{\mathbb{R}}^{m\times n}$. If $Q$ has exactly one positive
eigenvalue, then $PQP^{t}$ has at most one positive eigenvalue.

######  Remark 4.3.

Let $Q$ be a symmetric matrix. Then $Q$ has exactly one positive eigenvalue if
and only if $Q$ has at most one positive eigenvalue and there exists
$a\in{\mathbb{R}}^{n}$ such that $a^{t}Qa>0$.

The following lemma can be found in [HW17] and [AGV21]. However, for the sake
of completeness, we provide a proof here to keep the paper self-contained.

######  Lemma 4.4.

Let $Q$ be a symmetric matrix and $a\in{\mathbb{R}}^{n}$ such that
$a^{t}Qa>0$. Then the following statements are equivalent:

  1. (a)

$Q$ has exactly one positive eigenvalue

  2. (b)

the matrix $(a^{t}Qa)\cdot Q-t(Qa)(Qa)^{t}$ is negative semidefinite for all
$t\geq 1$.

  3. (c)

$Q$ is negative semidefinite on the hyperplane $(Qa)^{t}$.

###### Proof.

$(a\Rightarrow b)$ Since the matrix $(Qa)(Qa)^{t}$ is positive semidefinite,
it suffices to prove part (2) for $t=1$. Suppose $b\in{\mathbb{R}}^{n}$ is not
a scalar multiple of $a$. Consider the $2\times n$ matrix $P$ with rows
$a^{t}$ and $b^{t}$. Then

(1) $PQP^{t}=\begin{bmatrix}a^{t}Qa&a^{t}Qb\\\ b^{t}Qa&b^{t}Qb&\end{bmatrix}$

By Lemma 4.2, the matrix $PQP^{t}$ has at most one positive eigenvalue. Then,
$\det(PQP^{t})=a^{t}Qa\cdot b^{t}Qb-a^{t}Qb\cdot b^{t}Qa\leq 0$. Thus,

$b^{t}\left((a^{t}Qa)\cdot Q-(Qa)(Qa)^{t}\right)b\leq 0$

for all $b\in{\mathbb{R}}^{n}$.

($b\Rightarrow c$) Since $a^{t}Qa\cdot Q-(Qa)(Qa)^{t}$ is negative
semidefinite and $a^{t}Qa>0$, we have

$b^{t}\left((a^{t}Qa)\cdot Q-(Qa)(Qa)^{t}\right)b\leq 0$

for all $b\in{\mathbb{R}}^{n}$. Therefore $a^{t}Qa\cdot b^{t}Qb-a^{t}Qb\cdot
b^{t}Qa\leq 0$. Thus, $Q$ is negative semidefinite on the hyperplane
$\\{b\in{\mathbb{R}}^{n}\,\mid\,b^{t}Qa=0\\}=(Qa)^{\perp}$.

($c\Rightarrow a$) Since $Q$ is negative semidefinite on the hyperplane
$\\{b\in{\mathbb{R}}^{n}\,\mid\,b^{T}Qa=0\\}=(Qa)^{\perp}$ and $a^{t}Qa>0$, we
see that $Q$ has exactly one positive eigenvalue. ∎

We now present several equivalent conditions for log-concavity of a form
$f\in{\mathbb{R}}[x]_{n}^{d}$ on a proper convex cone $\mathcal{K}$, which are
quite similar to the conditions described in [AGV21] and [BH20] for the
nonnegative orthant.

######  Proposition 4.5.

Let $f\in{\mathbb{R}}[x]_{n}^{d}$ be a form of degree $d\geq 2$ and
${\mathcal{K}}\subset{\mathbb{R}}^{n}$ be a proper convex cone. Suppose that
$f(a)>0$ for all $a\in\operatorname{int}{\mathcal{K}}$, and let $Q_{a}$ denote
the Hessian matrix of $f$ evaluated at a: $Q_{a}=H_{f}(a)$. The following
statements are equivalent:

  1. (a)

$f$ is log-concave on ${\mathcal{K}}$.

  2. (b)

$Q_{a}$ has exactly one positive eigenvalue for all
$a\in\operatorname{int}{\mathcal{K}}$.

  3. (c)

The quadratic form $x^{t}Q_{a}x$ is negative semidefinite on the hyperplane
$(Q_{a}a)^{\perp}$ for all $a\in\operatorname{int}{\mathcal{K}}$.

  4. (d)

For all $a\in\operatorname{int}{\mathcal{K}}$, the matrix
$(a^{t}Q_{a}a)Q_{a}-(Q_{a}a)(Q_{a}a)^{t}$ is negative semidefinite.

###### Proof.

Euler’s Identity states that for a homogeneous polynomial $f$ of degree $d$,
$\langle\nabla f(x),x\rangle=\sum_{i=1}^{n}x_{i}\partial_{i}f(x)=d\cdot f(x)$.
Using this on $f$ and $\partial_{i}f$ we easily get the following identities:

$Q_{a}a=(d-1)\cdot\nabla f(a),\ a^{t}Q_{a}a=d(d-1)\cdot f(a).$

Additionally, the Hessian of $\log(f)$ at $x=a$ equals

(2) $H_{\log f}(a)=\left(\frac{f\cdot H_{f}-\nabla f\nabla
f^{t}}{f^{2}}\right)|_{{\mathbf{x}}={\mathbf{a}}}=d(d-1)\frac{a^{t}Q_{a}a\cdot
Q_{a}-\frac{d}{d-1}(Q_{a}a)(Q_{a}a)^{t}}{(a^{t}Qa)^{2}}.$

($a\Rightarrow b$) Since $f$ is a non-zero form which is log-concave on all
points of the interior of ${\mathcal{K}}$ we see that $f(a)>0$ for all
$a\in\operatorname{int}{\mathcal{K}}$, and the Hessian of $\log(f)$ at $a$ is
negative semidefinite. Since $a^{t}Q_{a}a>0$ we see from (2) that $Q_{a}$ has
at most $1$ positive eigenvalue. However, $Q_{a}$ cannot be negative definite,
since $a^{t}Q_{a}a>0$. Thus $Q_{a}$ has exactly one positive eigenvalue.

($b\Leftrightarrow c\Leftrightarrow d$) The proof follows by Lemma 4.4. ∎

We also have analogous conditions for strict log-concavity.

######  Proposition 4.6.

Let $f\in{\mathbb{R}}[x]_{n}^{d}$ be a form of degree $d\geq 2$ and
${\mathcal{K}}\subset{\mathbb{R}}^{n}$ be a proper convex cone. Suppose that
$f(a)>0$ for all $a\in{\mathcal{K}}$, and let $Q_{a}$ denote the Hessian
matrix of $f$ evaluated at a: $Q_{a}=H_{f}(a)$. The following statements are
equivalent:

  1. (a)

$f$ is strictly log-concave on ${\mathcal{K}}$.

  2. (b)

For all $a\in{\mathcal{K}}$, the quadratic form $x^{t}Q_{a}x$ is negative
definite on the hyperplane $(Q_{a}a)^{\perp}$ .

  3. (c)

The matrix $Q_{a}$ is nonsingular and has exactly one positive eigenvalue for
all $a\in{\mathcal{K}}$.

###### Proof.

($a\Rightarrow b$) The Hessian of $\log(f(x))$ at $x=a$ is negative definite.
Then by Equation (2) the matrix $M_{a}:=(a^{t}Q_{a}a)\cdot
Q_{a}-\frac{d}{d-1}(Q_{a}a)(Q_{a}a)^{t}$ is negative definite. Consider the
restriction of the quadratic form $x^{t}M_{a}x$ to the hyperplane
$(Q_{a}a)^{\perp}$. For $x$ in this hyperplane we have
$x^{t}M_{a}x=(a^{t}Q_{a}a)\cdot x^{t}Q_{a}x$, and so $x^{t}Q_{a}x$ is negative
definite on $(Q_{a}a)^{\perp}$.

($b\Rightarrow c$) Since $x^{t}Q_{a}x$ is negative definite on the hyperplane
$(Q_{a}a)^{\perp}$, we see that $Q$ has at least $n-1$ negative eigenvalues.
But, since $d(d-1)\cdot f(a)=a^{t}Q_{a}a>0$ we see that $Q_{a}$ must also have
a positive eigenvalue.

($c\Rightarrow a$) We need to establish that the matrix $a^{t}Q_{a}a\cdot
Q_{a}-\frac{d}{d-1}(Q_{a}a)(Q_{a}a)^{t}$ is negative definite using the fact
that $Q_{a}$ is nonsingular. The proof is exactly the same as the proof of
$(a\Rightarrow b)$ in Lemma 4.4.

∎

The following result shows that $f$ is log-concave at $a$ if and only if its
directional derivative in the direction $a$ is log-concave at $a$, cf.
[ALOGV24b, Lemma 2.1].

######  Proposition 4.7.

Let $f\in{\mathbb{R}}[x]_{n}^{d}$ be a homogeneous polynomial of degree $d\geq
3$ with $f(a)>0$ at some point $a\in{\mathbb{R}}^{n}$. Then $f$ is log-concave
at $a$ if and only if $D_{a}(f)$ is log-concave at $a$.

###### Proof.

By using Euler’s identity for a homogeneous polynomial of degree $d$, we can
express $d\cdot f(x)=\sum_{i=1}^{n}x_{i}\frac{\partial f}{\partial
x_{i}}=D_{x}(f(x))$. Therefore, $f(a)>0$ if and only if $D_{a}f(a)>0$.
Moreover, $H_{D_{a}f}(a)=(d-2)H_{f}(a)$. Then at the point $x=a$, signature of
$H_{f}(a)$ and $H_{D_{a}f}(a)$ are the same. Then the claim follows from
Proposition 4.5. ∎

We also show that log-concavity at a point is preserved under invertible
linear transformations.

######  Lemma 4.8.

If $f(x)\in{\mathbb{R}}[x]_{n}^{d}$ is log-concave at $a$, then $f(Ax)$ is
log-concave at $A^{-1}a$ for any nonsingular $n\times n$ matrix $A$.

###### Proof.

The Hessian of $f(Ax)$ at $A^{-1}a$ is equal to $A^{t}H_{f}(a)A$. Therefore
the Hessian of $f(Ax)$ at $A^{-1}a$ has exactly one positive eigenvalue and
log-concavity of $f(Ax)$ at $A^{-1}a$ follows by Proposition 4.5. ∎

###  4.2. Interior

Brändén and Leake [BL23] proved that
$\operatorname{\mathrm{SL}}^{d}({\mathcal{K}})$ is a closed set, and it is the
closure of its interior. We use their result to describe the interior of
$\operatorname{\mathrm{SL}}^{d}({\mathcal{K}})$. We define the set
$\operatorname{\mathring{SL}}^{d}({\mathcal{K}})$ as the set consisting forms
in $\operatorname{\mathrm{SL}}^{d}({\mathcal{K}})$ which additionally satisfy
the following properties: for all $a_{1},\dots,a_{d-2}\in{\mathcal{K}}$ the
matrix $Q$ of $q=D_{a_{1}}\dots D_{a_{d-2}}f$ is nonsingular and $x^{t}Qy>0$
for all $x,y\in{\mathcal{K}}$, i.e. strict positivity in Definition 2.2 holds
on all of ${\mathcal{K}}$, and not just in the interior of ${\mathcal{K}}$.
Next we show that $\operatorname{\mathring{SL}}^{d}({\mathcal{K}})$ is the
interior of $\operatorname{\mathrm{SL}}^{d}({\mathcal{K}})$.

######  Theorem 4.9.

Let ${\mathcal{K}}\subset{\mathbb{R}}^{n}$ be a proper convex cone. Then
$\operatorname{\mathring{SL}}^{d}_{n}({\mathcal{K}})$ is the interior of
$\operatorname{\mathrm{SL}}^{d}_{n}({\mathcal{K}})$, and

$\displaystyle\operatorname{\mathrm{SL}}^{d}_{n}({\mathcal{K}})=\overline{\operatorname{\mathring{SL}}^{d}_{n}({\mathcal{K}})}$

###### Proof.

It suffices to show that (1) the set
$\operatorname{\mathring{SL}}^{d}_{n}({\mathcal{K}})$ is open, and (2) any
form
$f\in\operatorname{\mathrm{SL}}^{d}_{n}({\mathcal{K}})\setminus\operatorname{\mathring{SL}}^{d}_{n}({\mathcal{K}})$
does not lie in the interior of
$\operatorname{\mathrm{SL}}^{d}_{n}({\mathcal{K}})$.

Openness of $\operatorname{\mathrm{SL}}^{d}({\mathcal{K}})$ is clear for
$d\leq 1$. For $d\geq 2$, let
$f\in\operatorname{\mathring{SL}}^{d}({\mathcal{K}})$ and let $B$ be the
intersection of ${\mathcal{K}}$ with the unit sphere in ${\mathbb{R}}^{n}$:
$B=\mathbb{S}^{n-1}\cap{\mathcal{K}}$. Let $\alpha$ be the infimum of
$x^{t}Qy$ where $q=D_{a_{1}}\cdots D_{a_{d-2}}f$ with $a_{i},x,y\in B$. Since
we are taking the infimum of a continuous function on a compact set $B^{d}$
($d-2$ choices for $a_{i}$ and one each for $x$ and $y$), we see that
$\alpha>0$. Otherwise the minimum of $0$ is attained on some point of $B^{d}$,
which contradicts $f\in\operatorname{\mathring{SL}}^{d}({\mathcal{K}})$.
Similarly let $\beta$ be the minimum of $|\det Q|$ over all choices of
$a_{1},\dots,a_{d-2}\in B$. Again we see that $\beta>0$. Therefore for any
$g\in{\mathbb{R}}[x]_{n}^{d}$ there exists $\varepsilon>0$ such that
$f+\varepsilon g\in\operatorname{\mathring{SL}}^{d}({\mathcal{K}})$. Thus we
see that $\operatorname{\mathring{SL}}^{d}({\mathcal{K}})$ is open.

If a form $f$ lies in
$\operatorname{\mathrm{SL}}^{d}({\mathcal{K}})\setminus\operatorname{\mathring{SL}}^{d}({\mathcal{K}})$,
then for some choice of $a_{1},\dots,a_{d-2},x,y\in B$ we have $x^{t}Qy=0$, or
$Q$ is singular, where $q=D_{a_{1}}\cdots D_{a_{d-2}}f$. In the first case,
take $g\in{\mathbb{R}}[x]_{n}^{d}$ such that $x^{t}Q^{\prime}y<0$ where
$q^{\prime}=D_{a_{1}}\cdots D_{a_{d-2}}g$. Then for any $\varepsilon>0$ we
have $f+\varepsilon g\notin\operatorname{\mathrm{SL}}^{d}({\mathcal{K}})$, and
thus $f\notin\operatorname{int}\operatorname{\mathrm{SL}}^{d}({\mathcal{K}})$.
In the second case take $g\in{\mathbb{R}}[x]_{n}^{d}$ such that
$q^{\prime}=D_{a_{1}}\cdots D_{a_{d-2}}g$ is strictly positive on the kernel
of $Q$. Then for sufficiently small $\varepsilon>0$ we have $f+\varepsilon
g\notin\operatorname{\mathrm{SL}}^{d}({\mathcal{K}})$, and thus
$f\notin\operatorname{int}\operatorname{\mathrm{SL}}^{d}({\mathcal{K}})$. ∎

###  4.3. Equivalence of ${\mathcal{K}}$-Lorentzian and ${\mathcal{K}}$-CLC
forms

In this subsection, we will often utilize the equivalent defintion of
${\mathcal{K}}$-Lorentzian polynomials using positivity of $D_{a_{1}}\dots
D_{a_{d}}f$ for all $a_{1},\dots,a_{d}\in\operatorname{int}{\mathcal{K}}$, as
explained in Remark 2.3.

######  Theorem 4.10.

Let ${\mathcal{K}}\subset{\mathbb{R}}^{n}$ be a proper cone. A form
$f\in{\mathbb{R}}[x]_{n}^{d}$ is ${\mathcal{K}}$-Lorentzian if and only if $f$
is ${\mathcal{K}}$-CLC. Equivalently,
$\operatorname{\mathrm{SL}}^{d}({\mathcal{K}})=\mathrm{CLC}^{d}({\mathcal{K}})$.

###### Proof.

$\Leftarrow$ For $f\in\mathrm{CLC}^{d}({\mathcal{K}})$, the form $f$ and all
its repeated directional derivatives are log-concave on
$\operatorname{int}{\mathcal{K}}$. Thus we see that, the conditions for a
polynomial to lie in $\operatorname{\mathrm{SL}}^{d}({\mathcal{K}})$ are a
subset of the conditions for a polynomial to lie in
$\mathrm{CLC}^{d}({\mathcal{K}})$.

$\Rightarrow$ According to Definition 2.2, nonnegative constants and linear
forms nonnegative on ${\mathcal{K}}$ are ${\mathcal{K}}$-Lorentzian
polynomials. We establish this result by induction on the degree $d$ of the
forms. For the base case $d=2$, let
$f\in\operatorname{\mathrm{SL}}^{2}({\mathcal{K}})$. The positivity condition
(2) in Definition 2.2 and Remark 2.3 ensure log-concavity for directional
derivatives of degree $1$ and $0$ for $f$. The signature condition (1) in
Definition 2.2 implies that $f$ is log-concave on
$\operatorname{int}{\mathcal{K}}$ by Proposition 4.5.

Now we assume that the result holds for forms in
$\operatorname{\mathrm{SL}}^{d-1}({\mathcal{K}})$. By the induction
hypothesis, $D_{a}f$ is ${\mathcal{K}}$-CLC for any
$a\in\operatorname{int}{\mathcal{K}}$, and since the set
$\operatorname{\mathrm{SL}}^{d-1}({\mathcal{K}})=\operatorname{\mathrm{CLC}}^{d-1}({\mathcal{K}})$
is closed, we see that $D_{a}f$ is ${\mathcal{K}}$-CLC for any
$a\in{\mathcal{K}}$. Thus it suffices to show that $f$ is log-concave on
$\operatorname{int}{\mathcal{K}}$. Similarly to the base case, the positivity
condition (2) in Definition 2.2 implies $f(a)>0$ for all
$a\in\operatorname{int}{\mathcal{K}}$. By the induction hypothesis, $D_{a}f$
is log-concave at any $a\in\operatorname{int}{\mathcal{K}}$, and we also have
$H_{D_{a}f}(a)=(d-2)H_{f}(a)$. Therefore, by Proposition 4.7 we see that $f$
is log-concave at $a$. Thus,
$f\in\operatorname{\mathrm{CLC}}^{d}({\mathcal{K}})$. ∎

######  Remark 4.11.

By Theorem 4.10 and Theorem 4.9 we have shown that
$\operatorname{\mathrm{CLC}}^{d}({\mathcal{K}})$ is the closure of its
interior, denoted as
$\operatorname{int}\operatorname{\mathrm{CLC}}^{d}({\mathcal{K}})$, consisting
of strictly ${\mathcal{K}}$-CLC polynomials.

###  4.4. ${\mathcal{K}}$-CLC forms over proper convex cones

We now derive some consequences for the structure of the set
$\operatorname{\mathrm{CLC}}^{d}({\mathcal{K}})=\operatorname{\mathrm{SL}}^{d}({\mathcal{K}})$.
We use $H_{f}$ to denote the Hessian matrix of a form $f$.

######  Lemma 4.12.

Let $f\in{\mathbb{R}}[x]^{d}_{n}$ be a nonzero polynomial and ${\mathcal{K}}$
be a proper convex cone.

  1. (a)

If $f\in\operatorname{\mathrm{CLC}}^{d}({\mathcal{K}})$, then $H_{f}(a)$ has
exactly one positive eigenvalue for all $a\in\operatorname{int}{\mathcal{K}}$.

  2. (b)

If $f\in\operatorname{int}\operatorname{\mathrm{CLC}}^{d}({\mathcal{K}})$,
then $H_{f}(a)$ is nonsingular and has exactly one positive eigenvalue for all
$a\in{\mathcal{K}}$.

###### Proof.

$(a)$ follows from Proposition 4.5.
($b$) $f\in\operatorname{int}\operatorname{\mathrm{CLC}}^{d}({\mathcal{K}})$
implies that $f$ is strictly log-concave on all points of ${\mathcal{K}}$. The
rest follows from Proposition 4.6. ∎

######  Proposition 4.13.

Let $f\in{\mathbb{R}}[x]_{n}^{d}$ be a nonzero form of degree $d\geq 2$,
${\mathcal{K}}$ be a proper convex cone in ${\mathbb{R}}^{n}$, and
$Q_{a}=H_{f}(a)$.

  1. (a)

Let $f\in\operatorname{int}\operatorname{\mathrm{CLC}}^{d}({\mathcal{K}})$.
Then quadratic form $x^{t}Q_{a}x$ is negative definite on $(Q_{a}b)^{\perp}$
for every $a,b\in{\mathcal{K}}$ such that $Q_{a}b\neq 0$.

  2. (b)

Let $f\in\operatorname{\mathrm{CLC}}^{d}({\mathcal{K}})$. Then quadratic form
$x^{t}Q_{a}x$ is negative semidefinite on $(Q_{a}b)^{\perp}$ for every
$a\in\operatorname{int}{\mathcal{K}}$, and $b\in{\mathcal{K}}$ such that
$Q_{a}b\neq 0$.

###### Proof.

(a) Given that
$f\in\operatorname{int}\operatorname{\mathrm{CLC}}^{d}({\mathcal{K}})$ and
$a\in{\mathcal{K}}$, we know that $Q_{a}$ is nonsingular has exactly one
positive eigenvalue by Lemma 4.12. Additionally, since $b\in{\mathcal{K}}$, so
it follows from the Definition 2.1 that $b^{t}Q_{a}b=D_{b}D_{b}f(a)>0$. Then
the result follows from Lemma 4.4.
(b) This follows from (a) by taking limits and using Remark 4.11. ∎

######  Lemma 4.14.

Let $f\in{\mathbb{R}}[x]_{n}^{d}$ be a nonzero polynomial, ${\mathcal{K}}$ be
a proper convex cone in ${\mathbb{R}}^{n}$, and $Q_{a}:=H_{f}(a)$ for
$a\in{\mathcal{K}}$.

  1. (a)

If $f\in\operatorname{int}\operatorname{\mathrm{CLC}}^{d}({\mathcal{K}})$, the
restriction of $Q_{a}$ on a plane spanned by $b,c\in{\mathcal{K}}$ is a
positive matrix with negative determinant.

  2. (b)

If $f\in\operatorname{\mathrm{CLC}}^{d}({\mathcal{K}})$, the restriction of
$Q_{a}$ on a plane spanned by $b,c\in{\mathcal{K}}$ is a nonnegative matrix
with nonpositive determinant.

###### Proof.

$(a)$ By Lemma 4.12, $Q:=H_{f}(a)$ is nonsingular and has exactly one positive
eigenvalue. Then the rest follows from Remark 2.3 and the restriction of the
Hessian to the plane spanned by $b,c\in{\mathcal{K}}$, outlined in (1).

$(b)$ Since $\operatorname{\mathrm{CLC}}^{d}({\mathcal{K}})$ is the closure of
its interior (See Remark 4.11), the result follows from part $(a)$ by taking
limits. ∎

###  4.5. The sum of two Lorentzian polynomials

The sum of two log-concave polynomials may not be log-concave. The subsequent
result provides a sufficient condition for the sum of two
${\mathcal{K}}$-Lorentzian polynomials to also be ${\mathcal{K}}$-Lorentzian.
This condition is analogous to the one provided in [ALOGV24b, Lemma 3.3] for
the sum of two log-concave polynomials with nonnegative coefficients to be
log-concave on the nonnegative orthant.

######  Theorem 4.15.

Let $f,g\in\operatorname{\mathrm{SL}}^{d}({\mathcal{K}})$. If there exist
vectors $b,c\in{\mathcal{K}}$ such that $D_{b}f=D_{c}g\not\equiv 0$, then
$f+g$ is ${\mathcal{K}}$-Lorentzian.

###### Proof.

We prove this result by induction on the degree $d$. If $d=1$, then
$(f+g)(a)=f(a)+g(a)>0$ for any $a\in\operatorname{int}{\mathcal{K}}$. Thus,
$f+g$ is ${\mathcal{K}}$-Lorentzian.

Now, suppose $d\geq 2$ and fix $a\in\operatorname{int}{\mathcal{K}}$. Let
$f_{1}=D_{a}f$ and $g_{1}=D_{a}g$. Note that $f_{1}$ and $g_{1}$ are
${\mathcal{K}}$-Lorentzian polynomials, and additionally
$D_{b}f_{1}=D_{b}D_{a}f=D_{a}D_{b}f=D_{a}D_{c}g=D_{c}D_{a}g=D_{c}g_{1}$. If we
can show that $D_{b}f_{1}=D_{c}g_{1}\not\equiv 0$ for all
$a\in\operatorname{int}{\mathcal{K}}$, then by the induction assumption we
have $f_{1}+g_{1}=D_{a}(f+g)$ is ${\mathcal{K}}$-Lorentzian for all
$a\in\operatorname{int}{\mathcal{K}}$ and therefore $f+g$ is
${\mathcal{K}}$-Lorentzian.

Since the set $\operatorname{\mathrm{SL}}^{d-1}({\mathcal{K}})$ is closed it
follows that $D_{b}f$ is a nonzero polynomial in
$\operatorname{\mathrm{SL}}^{d-1}({\mathcal{K}})$. In particular, $D_{b}f$ is
strictly positive on $\operatorname{int}K$, and therefore
$D_{a}D_{b}f(a)=(d-1)(D_{b}f)(a)>0$. Thus we see that
$D_{b}f_{1}=D_{b}D_{a}f=D_{a}D_{b}f$ is not identically zero. ∎

##  5\. ${\mathcal{K}}$-Lorentzian polynomials over self-dual cones

In this section, we outline an interesting connection
${\mathcal{K}}$-Lorentzian polynomials on a self-dual cone ${\mathcal{K}}$ and
the generalized Perron-Frobenius theorem. We refer the reader to [GL23] for
various classes of self-dual cones other than the nonnegative orthant, the
second order cone, and the cone of positive semidefnite matrices.

###  5.1. ${\mathcal{K}}$-irreducibility

We now show an alternative characterization of the set of Lorentzian
polynomials $\operatorname{\mathrm{SL}}^{d}({\mathcal{K}})$ for a self-dual
cone ${\mathcal{K}}$. We note that for a quadratic form $q$, the Hessian
matrix $H_{q}$ is equal to twice the matrix $Q$ of $q$. Therefore our results
can be equivalently stated in terms of the matrix $H_{q}$.

######  Theorem 5.1.

Let $f\in{\mathbb{R}}[x]_{n}^{d}$ be a form of degree $d\geq 2$ and
${\mathcal{K}}$ be a self-dual cone. Then
$f\in\operatorname{\mathrm{SL}}^{d}({\mathcal{K}})$ if and only if for all
$a_{1},\dots,a_{d-2}\in\operatorname{int}{\mathcal{K}}$ the matrix $Q$ of the
quadratic form $q=D_{a_{1}}\dots D_{a_{d-2}}f$ satisfies the following two
conditions:

  1. (a)

$Q$ either nonsingular and ${\mathcal{K}}$-irreducible, or singular and
${\mathcal{K}}$-nonnegative.

  2. (b)

$Q$ has exactly one positive eigenvalue.

###### Proof.

This follows from Definition 2.2 and Corollary 3.10. ∎

Therefore, we have the following characterization of Lorentzian polynomials
or, equivalently, CLC polynomials on the nonnegative orthant.

######  Corollary 5.2.

Let $f\in{\mathbb{R}}[x]_{n}^{d}$ be a form of degree $d\geq 2$. Then $f$ is
Lorentzian or CLC polynomial on ${\mathbb{R}}^{n}_{\geq 0}$ if and only if for
all $a_{1},\dots,a_{d-2}\in{\mathbb{R}}^{n}_{>0}$, the matirx $Q$ of the
quadratic form $q=D_{a_{1}}\dots D_{a_{d-2}}f$ satisfies the following two
conditions:

  1. (1)

$Q$ is either nonsingular and irreducible, or singular and nonnegative.

  2. (2)

$Q$ has exactly one positive eigenvalue.

We now provide an alternative characterization of the interior
$\operatorname{\mathring{SL}}^{d}({\mathcal{K}})$ of the set of
${\mathcal{K}}$-Lorentzian polynomials on a self-dual cone ${\mathcal{K}}$
using ${\mathcal{K}}$-positivity.

######  Theorem 5.3.

Let $f\in{\mathbb{R}}[x]_{n}^{d}$ be a homogeneous polynomial of degree $d\geq
2$ and ${\mathcal{K}}$ be a self-dual cone. Then
$f\in\operatorname{\mathring{SL}}^{d}({\mathcal{K}})$ if and only if for any
$a_{1},\dots,a_{d-2}\in{\mathcal{K}}$, the matrix $Q$ of the quadratic form
$q=D_{a_{1}}\dots D_{a_{d-2}}f$ satisfies the following two conditions:

  1. (1)

$Q$ is ${\mathcal{K}}$-positive and nonsingular.

  2. (2)

$Q$ has exactly one positive eigenvalue.

###### Proof.

This follows from Theorem 4.9 (see also observations in Lemma 3.1 and
Corollary 3.8). ∎

###  5.2. Extreme Rays

Let ${\mathcal{K}}\subseteq{\mathbb{R}}^{n}$ be a self-dual cone and
$f\in{\mathbb{R}}[x]_{n}^{d}$ be a homogeneous polynomial of degree $d$ in $n$
variables. By the conical Minkowski theorem, a proper convex cone
${\mathcal{K}}\subseteq{\mathbb{R}}^{n}$ is generated by its extreme rays. We
recall that ${\mathcal{K}}$-CLC polynomials are a closed set in
${\mathbb{R}}_{n}^{d}[x]$, which is the closure of its interior, which
consists of strictly ${\mathcal{K}}$-CLC polynomials. Therefore, we may ask
whether taking directional derivatives with respect to extreme rays is
sufficient to guarantee that a polynomial is ${\mathcal{K}}$-CLC. First we
demonstrate that in general, testing ${\mathcal{K}}$-CLC conditions on extreme
rays of ${\mathcal{K}}$ is not sufficient.

######  Example 5.4.

Consider the polynomial $f=x_{1}(x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3})+x_{4}^{3}$.
Then over the nonnegative orthant $D_{e_{i}}f$ are ${\mathcal{K}}$-Lorentzian
polynomials, but $D_{{\mathbf{1}}}f$ is not where ${\mathbf{1}}$ denotes all-
ones vector. Consequently, $f$ is not a ${\mathcal{K}}$-Lorentzian polynomial
over the nonnegative orthant.

However, with additional assumptions on the polynomial
$f\in{\mathbb{R}}[x]_{n}^{d}$, testing ${\mathcal{K}}$-CLC conditions on the
extreme rays of a self-dual cone ${\mathcal{K}}$ will be sufficient.

######  Lemma 5.5.

Let $f\in{\mathbb{R}}[x]_{n}^{d}$ be a form of degree $d\geq 2$. Let
${\mathcal{K}}\subset{\mathbb{R}}^{n}$ be a self-dual cone and
$E_{{\mathcal{K}}}$ denote the set of extreme rays of cone ${\mathcal{K}}$. If
the Hessian matrix $H_{f}(v)$ is ${\mathcal{K}}$-positive for all $v\in
E_{{\mathcal{K}}}$, then $D_{v_{1}}D_{v_{2}}f$ is not identically zero for all
$v_{1},v_{2}\in E_{{\mathcal{K}}}$.

###### Proof.

If $D_{v_{1}}D_{v_{2}}f\equiv 0$ then we have $v_{2}^{t}H_{f}(x)v_{1}=0$ for
all $x\in{\mathbb{R}}^{n}$. This contradicts ${\mathcal{K}}$-positivity of
$H_{f}(v_{i})$. ∎

######  Remark 5.6.

The above result does not hold if we only require that $H_{f}(v)$ is
${\mathcal{K}}$-irreducible for all $v\in E_{k}$. For instance, consider
$f=x_{1}x_{2}+x_{1}x_{3}$ with ${\mathcal{K}}$ the nonnegative orthant in
${\mathbb{R}}^{3}$. While the Hessian matrices are nonsingular and
irreducible, we have $\partial_{2}\partial_{3}f$ is identically zero.

######  Theorem 5.7.

Let $f\in{\mathbb{R}}[x]_{n}^{d}$ be a form of degree $d\geq 3$, and
${\mathcal{K}}$ be a self-dual cone where $E_{{\mathcal{K}}}$ denotes the set
of its extreme rays. Suppose that the Hessian $H_{f}(v)$ is nonsingular and
${\mathcal{K}}$-positive for all $v\in E_{{\mathcal{K}}}$. If $D_{v}f$ is
${\mathcal{K}}$-CLC for all $v\in E_{{\mathcal{K}}}$, then $f$ is
${\mathcal{K}}$-CLC.

###### Proof.

Let $a\in{\mathcal{K}}$ and write $a=\sum_{i=1}^{m}c_{i}v_{i}$ for real
positive $c_{i}$ and $v_{i}\in E_{\mathcal{K}}$. Then we have
$D_{a}f=\sum_{i=1}^{m}c_{i}D_{v_{i}}f$, and
$D_{v_{j}}D_{v_{i}}f=D_{v_{i}}D_{v_{j}}f$. Now by Lemma 5.5 all
$D_{v_{j}}D_{v_{i}}f$ are nonzero polynomials and we can apply Theorem 4.15 to
see that $D_{a}f$ is ${\mathcal{K}}$-CLC for all $a\in{\mathcal{K}}$. This
implies that $f$ is ${\mathcal{K}}$-CLC by the equivalence between
${\mathcal{K}}$-Lorentzian and ${\mathcal{K}}$-CLC polynomials. ∎

Finally, we provide an alternative proof of another necessary and sufficient
condition first proved in [BL23, Theorem 8.15] for
$f\in{\mathbb{R}}[x]_{n}^{d}$ to be a ${\mathcal{K}}$-Lorentzian polynomial.
We demonstrate that testing conditions in Definition 2.2 for directional
derivatives along the extreme rays and one more additional point in the
interior actually suffices.

######  Lemma 5.8.

Let $f\in{\mathbb{R}}[x]_{n}^{d}$ be a form of degree $d\geq 2$, and let
${\mathcal{K}}$ be a proper convex cone where $E_{{\mathcal{K}}}$ denotes the
set of its extreme rays, and $a\in\operatorname{int}{\mathcal{K}}$. If
$D_{a}f$ and $D_{v_{i}}f$ are ${\mathcal{K}}$-CLC, then the sum $D_{a+v}f$ for
any $v\in E_{{\mathcal{K}}}$ is also ${\mathcal{K}}$-CLC.

###### Proof.

We observe that for any non-zero ${\mathcal{K}}$-CLC polynomial $g$ we have
that $D_{a}g$ is a nonzero polynomial for all
$a\in\operatorname{int}{\mathcal{K}}$. If $D_{v}f$ is identically zero, then
the statement follows, so we may assume that $D_{v}f$ is a nonzero polynomial.
Since $D_{v}f$ is ${\mathcal{K}}$-CLC, we see that $D_{v}D_{a}f=D_{a}D_{v}f$
is a nonzero polynomial. We now conclude that the sum $D_{v+a}f=D_{v}f+D_{a}f$
is ${\mathcal{K}}$-CLC by Theorem 4.15. ∎

######  Theorem 5.9.

Let $f\in{\mathbb{R}}[x]_{n}^{d}$ be a form of degree $d\geq 2$, and let
${\mathcal{K}}$ be a proper convex cone where $E_{{\mathcal{K}}}$ denotes the
set of its extreme rays, and $a\in\operatorname{int}{\mathcal{K}}$. Then $f$
is ${\mathcal{K}}$-Lorentzian if and only if for all $v_{1},\dots,v_{d-2}\in
E_{{\mathcal{K}}}$ and all $0\leq k\leq d-2$, the quadratic form
$D_{v_{1}}D_{v_{2}}\dots D_{v_{k}}D_{a}^{d-2-k}f$ is
${\mathcal{K}}$-Lorentzian.

###### Proof.

$\Rightarrow$ Since $\operatorname{\mathrm{SL}}^{m}({\mathcal{K}})$ is closed
for all $m\geq 0$ we see that conditions on directional derivatives in
directions of extreme rays and $a$ are just a subset of conditions defining
${\mathcal{K}}$-Lorentzian polynomials.

$\Leftarrow$ We induct on the degree $d$ of $f$. The base case $d=2$ is
trivially true since we do not take any directional derivatives.

For $d\geq 3$ we assume that the result is true up to degree $d-1$. It
suffices to show that
$D_{b}f\in\operatorname{\mathrm{SL}}^{d-1}({\mathcal{K}})$ for all
$b\in\operatorname{int}{\mathcal{K}}$. We can write any
$b\in\operatorname{int}{\mathcal{K}}$ as a positive linear combination of $a$
and extreme rays $v_{i}$, where the coeficient of $a$ is strictly positive:
$b=\lambda a+\sum_{i=1}^{m}c_{i}v_{i}$ with $\lambda,c_{i}>0$, and $v_{i}\in
E_{{\mathcal{K}}}$.

Now we iteratively apply Lemma 5.8 to $\lambda a+c_{1}v_{1}$, $(\lambda
a+c_{1}v_{1})+c_{2}v_{2}$, $(\lambda a+c_{1}v_{1}+c_{2}v_{2})+c_{3}v_{3}$ and
so on, since $\lambda a+\sum c_{i}v_{i}$ always lies in the interior of
${\mathcal{K}}$.

∎

An immediate corollary provides a characterization of Lorentzian polynomials
which is first proved in [BL23, Corollary 8.16].

######  Corollary 5.10.

Let $f\in{\mathbb{R}}[x]_{n}^{d}$ be a form of degree $d\geq 2$ and
${\mathbf{1}}$ denote the all-ones vector. Then $f$ is Lorentzian if and only
if for all $i_{1},\dots,i_{d-2}\in[n]$ and all $0\leq k\leq d-2$, the
quadratic polynomial
$\partial_{i_{1}}\dots\partial_{i_{k}}D^{d-2-k}_{{\mathbf{1}}}f$ is
Lorentzian.

For a self-dual cone ${\mathcal{K}}$, using Theorem 5.1, we have the following
result.

######  Theorem 5.11.

Let $f\in{\mathbb{R}}[x]_{n}^{d}$ be a form of degree $d\geq 2$ and
${\mathcal{K}}$ be a self-dual cone where $E_{{\mathcal{K}}}$ denotes the set
of its extreme rays. Fix $a\in\operatorname{int}{\mathcal{K}}$. Then $f$ is
${\mathcal{K}}$-Lorentzian if and only if for all $v_{1},\dots,v_{k}\in
E_{{\mathcal{K}}}$ where $0\leq k\leq d-2$, the matrix $Q$ of the quadratic
form $q=D_{v_{1}}D_{v_{2}}\dots D_{v_{k}}D_{a}^{d-2-k}f$ satisfies the
following two conditions:

  1. (1)

$Q$ either nonsingular and ${\mathcal{K}}$-irreducible, or singular and
${\mathcal{K}}$-nonnegative.

  2. (2)

$Q$ has exactly one positive eigenvalue.

As a consequence we provide another alternative necessary and sufficient
conditions for Lorentzian polynomials.

######  Corollary 5.12.

Let ${\mathbf{1}}\in{\mathbb{R}}^{n}_{>0}$ denote the all-ones vector. $f$ is
Lorentzian over ${\mathbb{R}}^{n}_{\geq 0}$ if and only if for all
$i_{1},\dots,i_{d-2}\in[n]$ and $a\in{\mathbb{R}}^{n}_{>0},0\leq k\leq d-2$,
the matrix $Q$ of the quadratic form
$q=\partial_{i_{1}}\dots\partial_{i_{k}}D_{{\mathbf{1}}}^{d-2-k}f$ satisfies
the following two conditions:

  1. (1)

$Q$ is either nonsingular and irreducible, or singular and nonnegative.

  2. (2)

$Q$ has exactly one positive eigenvalue.

Note that it is easy to check irreducibility of a nonnegative matrix due to
the following result, see [Bru79].

######  Theorem 5.13.

An $n\times n$ nonnegative matrix $A$ is permutation equivalent to an
irreducible matrix if and only if $A$ contains no zero row or zero column.

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