# Differential Sensitivity of the KM3NeT/ARCA detector to a diffuse neutrino
flux and to point-like source emission: exploring the case of the Starburst
Galaxies

S. Aiello  A. Albert  M. Alshamsi  S. Alves Garre  Z. Aly  A. Ambrosone
<EMAIL_ADDRESS>F. Ameli  M. Andre  E. Androutsou  M. Anguita  L.
Aphecetche  M. Ardid  S. Ardid  H. Atmani  J. Aublin  F. Badaracco  L. Bailly-
Salins  Z. Bardačová  B. Baret  A. Bariego-Quintana  S. Basegmez du Pree  Y.
Becherini  M. Bendahman  F. Benfenati  M. Benhassi  D. M. Benoit  E. Berbee
V. Bertin  S. Biagi  M. Boettcher  D. Bonanno  J. Boumaaza  M. Bouta  M.
Bouwhuis  C. Bozza  R. M. Bozza  H.Brânzaş  F. Bretaudeau  M. Breuhaus  R.
Bruijn  J. Brunner  R. Bruno  E. Buis  R. Buompane  J. Busto  B. Caiffi  D.
Calvo  S. Campion  A. Capone  F. Carenini  V. Carretero  T. Cartraud  P.
Castaldi  V. Cecchini  S. Celli  L. Cerisy  M. Chabab  M. Chadolias  A. Chen
S. Cherubini  T. Chiarusi  M. Circella  R. Cocimano  J. A. B. Coelho  A.
Coleiro  R. Coniglione  P. Coyle  A. Creusot  G. Cuttone  R. Dallier  Y.
Darras  A. De Benedittis  B. De Martino  V. Decoene  R. Del Burgo  I. Del
Rosso  L. S. Di Mauro  I. Di Palma  A. F. Díaz  C. Diaz  D. Diego-Tortosa  C.
Distefano  A. Domi  C. Donzaud  D. Dornic  M. Dörr  E. Drakopoulou  D. Drouhin
R. Dvornický  T. Eberl  E. Eckerová  A. Eddymaoui  T. van Eeden  M. Eff  D.
van Eijk  I. El Bojaddaini  S. El Hedri  A. Enzenhöfer  G. Ferrara  M. D.
Filipović  F. Filippini  D. Franciotti  L. A. Fusco  J. Gabriel  S.
Gagliardini  T. Gal  J. García Méndez  A. Garcia Soto  C. Gatius Oliver  N.
Geißelbrecht  H. Ghaddari  L. Gialanella  B. K. Gibson  E. Giorgio  I. Goos
P. Goswami  D. Goupilliere  S. R. Gozzini  R. Gracia  K. Graf  C. Guidi  B.
Guillon  M. Gutiérrez  H. van Haren  A. Heijboer  A. Hekalo  L. Hennig  J. J.
Hernández-Rey  W. Idrissi Ibnsalih<EMAIL_ADDRESS>G.
Illuminati  M. de Jong  P. de Jong  B. J. Jung  P. Kalaczyński  O. Kalekin  U.
F. Katz  G. Kistauri  C. Kopper  A. Kouchner  V. Kueviakoe  V. Kulikovskiy  R.
Kvatadze  M. Labalme  R. Lahmann  G. Larosa  C. Lastoria  A. Lazo  S. Le Stum
G. Lehaut  E. Leonora  N. Lessing  G. Levi  M. Lindsey Clark  F. Longhitano
F. Magnani  J. Majumdar  L. Malerba  F. Mamedov  J. Mańczak  A. Manfreda  M.
Manzaneda  M. Marconi  A. Margiotta<EMAIL_ADDRESS>A. Marinelli
<EMAIL_ADDRESS>C. Markou  L. Martin  J. A. Martínez-Mora  F.
Marzaioli  M. Mastrodicasa  S. Mastroianni  S. Miccichè  G. Miele  P.
Migliozzi  E. Migneco  M. L. Mitsou  C. M. Mollo  L. Morales-Gallegos  M.
Morga  A. Moussa  I. Mozun Mateo  R. Muller  M. R. Musone  M. Musumeci  S.
Navas  A. Nayerhoda  C. A. Nicolau  B. Nkosi  B. Ó Fearraigh  V. Oliviero  A.
Orlando  E. Oukacha  D. Paesani  J. Palacios González  G. Papalashvili  V.
Parisi  E.J. Pastor Gomez  A. M. Păun  G. E. Păvălaş  S. Peña Martínez  M.
Perrin-Terrin  J. Perronnel  V. Pestel  R. Pestes  P. Piattelli  C. Poirè  V.
Popa  T. Pradier  J. Prado  S. Pulvirenti  G. Quéméner  C.A. Quiroz-Rangel  U.
Rahaman  N. Randazzo  R. Randriatoamanana  S. Razzaque  I. C. Rea  D. Real  G.
Riccobene  J. Robinson  A. Romanov  A. Šaina  F. Salesa Greus  D. F. E.
Samtleben  A. Sánchez Losa  S. Sanfilippo  M. Sanguineti  C. Santonastaso  D.
Santonocito  P. Sapienza  J. Schnabel  J. Schumann  H. M. Schutte  J. Seneca
N. Sennan  B. Setter  I. Sgura  R. Shanidze  A. Sharma  Y. Shitov  F. Šimkovic
A. Simonelli  A. Sinopoulou  M.V. Smirnov  B. Spisso  M. Spurio  D.
Stavropoulos  I. Štekl  M. Taiuti  Y. Tayalati  H. Thiersen  I. Tosta e Melo
E. Tragia  B. Trocmé  V. Tsourapis  A.Tudorache  E. Tzamariudaki  A. Vacheret
A. Valer Melchor  V. Valsecchi  V. Van Elewyck  G. Vannoye  G. Vasileiadis  F.
Vazquez de Sola  C. Verilhac  A. Veutro  S. Viola  D. Vivolo  J. Wilms  E. de
Wolf  H. Yepes-Ramirez  G. Zarpapis  S. Zavatarelli  A. Zegarelli  D. Zito  J.
D. Zornoza  J. Zúñiga  N. Zywucka  INFN, Sezione di Catania, (INFN-CT) Via
Santa Sofia 64, Catania, 95123 Italy  Université de Strasbourg, CNRS, IPHC UMR
7178, F-67000 Strasbourg, France  Aix Marseille Univ, CNRS/IN2P3, CPPM,
Marseille, France  University of Sharjah, Sharjah Academy for Astronomy, Space
Sciences, and Technology, University Campus - POB 27272, Sharjah, - United
Arab Emirates  IFIC - Instituto de Física Corpuscular (CSIC - Universitat de
València), c/Catedrático José Beltrán, 2, 46980 Paterna, Valencia, Spain
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Cintia ed. G, Napoli, 80126 Italy  Università di Napoli “Federico II”, Dip.
Scienze Fisiche “E. Pancini”, Complesso Universitario di Monte S. Angelo, Via
Cintia ed. G, Napoli, 80126 Italy  INFN, Sezione di Roma, Piazzale Aldo Moro
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Rambla Exposició, s/n, Vilanova i la Geltrú, 08800 Spain  NCSR Demokritos,
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15310 Greece  University of Granada, Dept. of Computer Architecture and
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Astroparticule et Cosmologie, F-75013 Paris, France  INFN, Sezione di Genova,
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81100 Italy  E. A. Milne Centre for Astrophysics, University of Hull, Hull,
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###### Abstract

KM3NeT/ARCA is a Cherenkov neutrino telescope under construction in the
Mediterranean sea, optimised for the detection of astrophysical neutrinos with
energies above $\sim$1 TeV. In this work, using Monte Carlo simulations
including all-flavour neutrinos, the integrated and differential sensitivities
for KM3NeT/ARCA are presented considering the case of a diffuse neutrino flux
as well as extended and point-like neutrino sources. This analysis is applied
to Starburst Galaxies demonstrating that the detector has the capability of
tracing TeV neutrinos from these sources. Remarkably, after eight years, a
hard power-law spectrum from the nearby Small Magellanic Cloud can be
constrained. The sensitivity and discovery potential for NGC 1068 is also
evaluated showing that KM3NeT/ARCA will discriminate between different
astrophysical components of the measured neutrino flux after 3 years of data
taking.

##  1 Introduction

The KM3NeT research infrastructure [1] is comprised of two deep-sea Cherenkov
neutrino telescopes under construction in the Mediterranean Sea: ARCA
(Astroparticle Research with Cosmics in the Abyss) located 3500 m below the
sea level, off shore the coast of Capo Passero (Sicily, Italy) and ORCA
(Oscillation Research with Cosmics in the Abyss) installed at a sea-bottom
depth of $\sim$2500 meters off shore Toulon (France). ARCA will have an
instrumented volume of about $1~{}\rm Km^{3}$ and its geometry is optimised to
detect high-energy astrophysical neutrinos above $\sim 1~{}\rm TeV$, while
ORCA is designed to study GeV neutrino physics analysing oscillation patterns
of atmospheric neutrinos. The detection principle for neutrinos111In this
paper, the word neutrino is used to refer to both neutrinos and antineutrinos
in KM3NeT is based on the observation of the Cherenkov light induced in the
sea water by secondary charged particles produced in the interaction of
neutrinos inside or in the surroundings of the detector. Given the excellent
optical seawater properties, the experiment will reach unprecedented angular
resolution in the identification of neutrino sources.

In this paper, the detection performance of the full ARCA telescope, both for
a diffuse neutrino flux and individual sources, are explored. Using Monte
Carlo simulations for all neutrino flavours, the energy-dependent 90%
confidence level (CL) sensitivity for ARCA is calculated, exploiting both
track-like and shower-like events. Track-like (referred to as tracks) events
are mainly due to charged current (CC) muon neutrino interactions; shower-like
(referred to as showers) events are mostly due to neutral current and electron
neutrinos charged current interactions.

For the track-like events, only upgoing events are considered, using the Earth
as a shield to reduce the atmospheric muon contamination. Shower-like events
are selected from the whole sky, using a containment requirement in the
fiducial volume of the detector in order to limit the contamination from
atmospheric muons and neutrinos. This selected sample is used to test if ARCA
is able to identify TeV neutrinos from Starburst Galaxies (SBGs), either as a
contribution to the diffuse flux or as emission from individual sources. SBGs
are galaxies experiencing intense phases of star formation activity, which
leads to an increased rate of supernova explosions. This enhanced activity is
expected to accelerate cosmic rays up to $\sim~{}\rm PeV$ energies and
copiously produce gamma-rays and neutrinos [2, 3]. In the diffuse flux
analysis, the SBG diffuse emission up to a redshift of $\sim~{}4$ is
considered [2]. In the individual source analysis, the Small Magellanic Cloud
(SMC), the Circinus Galaxy [3] and NGC1068 [4] have been analysed.

The paper is organised as follows: in Sec. 2, the event simulation and
selection are outlined. In Sec. 3, the statistical analysis framework is
presented. In particular, in Sec. 3.1, the definition of the sensitivity is
discussed, while in Sec. 3.2, the differential limits are introduced and
compared with the integrated limits.

In Secs. 4 and 5, the results of the analysis of the diffuse neutrino flux and
individual sources sensitivities are reported. In Sec. 6, the systematic
uncertainties on the sensitivity evaluation are discussed, while in Sec. 7
conclusions are reported. Further details are shown in the accompanying
appendices. In particular, A discusses the dependence of the sensitivity on
the source extension and finally in B, the variation of the sensitivity in
several declination bands is described.

##  2 Event Simulation and Selection

Neutrino event interactions in the proximity of the ARCA detector are
generated through the gSeaGen package [5]. The atmospheric muon flux is
simulated using the MUPAGE code [6]. Monte Carlo simulations continue
propagating charged particles emerging from neutrino interactions and
atmospheric muon tracks through the active volume of the detector and
reproducing the response of the front end electronics. The simulated data
stream is filtered and reconstructed with the same algorithms used for real
data. For this analysis, both upgoing tracks and all-sky showers samples are
used, following the event selection discussed in [7]. For the final selection,
two dedicated boosted decision trees are trained in order to reject mis-
reconstructed events in each sample. The neutrino purity of the two final
samples is $99\%$ for tracks and $61\%$ for showers. Moreover, the selection
preserves $95\%$ of the signal neutrinos for tracks and $70\%$ of the signal
neutrinos for showers [7]. The final background distribution accounts for
atmospheric neutrinos and muons.

##  3 Analysis Framework

###  3.1 Sensitivity Definition

In this paper, a binned maximum likelihood ratio method is used to evaluate
the ARCA sensitivity. Following the formalism and the notation of [8], the
likelihood function is defined as

$\displaystyle\mathcal{L}=\prod_{i}P(n_{i},\mu_{i})$ (1)
$\displaystyle\textrm{with}\,\mu_{i}=\lambda\cdot\mu_{s}^{i}+\mu_{b}^{i}$

where $P(n,\mu)$ is the Poisson probability distribution (PDF) of observing
$n$ events with mean value $\mu$, $\mu_{s}$ is the expected number of signal
events, while $\mu_{b}$ is the expected number of background events. The index
$i$ runs over bins of reconstructed variables. For the following, the events
are binned in reconstructed energy for the diffuse flux analysis and
reconstructed energy and angular distance of the events to the source for the
point-like analysis (see below for further details). The signal strength
$\lambda$ is left as a free normalisation parameter. Data samples are
simulated by means of pseudo-experiments (PEs) generation. The procedure
consists of randomly generating events according to PDFs of the signal and the
background and then evaluating the test statistic $(\rm TS)$ for each PE. The
$\rm TS$ is defined as

$\rm TS=\frac{\mathcal{L}(\tilde{\lambda})}{\mathcal{L}(\lambda=0)}$ (2)

where $\tilde{\lambda}$ is the signal strength value which maximises the
likelihood for a given PE. The $90\%$ CL sensitivity is defined as the signal
strength ($\lambda_{90}$) for which $90\%$ of the signal is above the median
of the background-only distribution. Namely,

$\int_{\rm TS_{m}}^{+\infty}d(\rm{TS}|\lambda_{90})~{}d\rm{TS}=90\%$ (3)

where $\rm TS_{m}$ is the median distribution in the null hypothesis (only
background) and $d(\rm{TS}|\lambda_{90})$ is the PDF distribution of the TS
for a given $\lambda_{90}$ value. Generally, $\lambda_{90}$ is referred to as
Model Rejection Factor (MRF), because, when signal simulations are performed
according to a theoretical model, a $\rm MRF\leq 1$ indicates an effective
$90\%$ CL constraint on that model. On the other hand, the model discovery
potential (MDP), also called the discovery flux, is defined as the minimum
flux needed for a discovery with $5\sigma$ significance level in $50\%$ of the
cases.

$\int_{\rm
TS_{5\sigma}}^{+\infty}d(\rm{TS}|\lambda_{5\sigma})~{}d\rm{TS}=50\%$ (4)

where $\rm TS_{5\sigma}=2.85\cdot 10^{-7}$ is the $\rm TS$ threshold
corresponding to a $5\sigma$ significance. This value is calculated using the
one-sided Gaussian approximation [9]. The final sensitivity is defined as

$\phi_{90}(E)=\lambda_{90}\phi_{s}(E)$ (5)

while the discovery flux as

$\phi_{5\sigma}(E)=\lambda_{5\sigma}\phi_{s}(E)$ (6)

where $\phi_{s}(E)$ is the signal flux injected in the simulations
corresponding to $\lambda=1$. Typically, the injected signal spectrum
$\phi_{s}(E)$ is modelled as a power-law $E^{-\gamma}$, resulting in
sensitivity and MDP profiles mirroring the shape of the injected spectrum. In
fact, Eqs. 5 and 6 provide results depending solely on the spectral shape and
are entirely independent of the specific scaling or normalisation of the
injected signal. Such expectations are also referred to as energy-integrated
expectations, see [7].

###  3.2 Differential Sensitivity and Discovery Flux

The differential sensitivity and discovery flux (in general, referred to as
‘differential limits’) correspond to the performance of a detector for a given
energy range [8]. In other words, they represent the minimum differential
signal neutrino flux that the detector can either constrain or discover. They
are only mildly model-dependent and represent important instrument response
functions. In order to evaluate them, the signal is divided in bins of
logarithm of true energy, injecting an $E^{-2}$ spectrum within each bin. The
width of the bins is chosen to be half decade in energy. Since the sensitivity
is inversely proportional to the number of signal events $n_{s}$ provided by
the injected spectrum [10], there is a strict connection between differential
and integrated sensitivities. In particular,

$\lambda_{90}\propto\rm
n_{s}^{-1}=\frac{1}{\sum_{i}n_{s}^{i}}\propto\frac{1}{\sum_{i}(\lambda_{90}^{i})^{-1}}$
(7)

where $\lambda_{90}$ is the integrated MRF, i runs over the number of energy
bins and $\lambda_{90}^{i}$ is the MRF for each energy bin of the signal.
The integrated MRF, as shown in Eq. 7, is always smaller than the differential
MRF. In fact, in differential limits, each individual energy bin is treated as
independent, assuming a zero signal flux outside the bin. This results in the
same amount of background events within each bin, leading to worse limits. The
integrated sensitivity exploits the energy dependence of the flux to
distinguish between the signal and the background, while the differential
sensitivity provides the generic differential flux which the detector can
discriminate, leading to a more stringent requirement. On the other hand, the
differential limits better highlight the energy range where the detector is
most sensitive, especially for different event samples (tracks and showers).
In the following, these quantities are computed and discussed in the context
of the sensitivity of ARCA to neutrino emission from starburst galaxies.

##  4 Diffuse Analysis

The $90\%$ CL differential sensitivity for a diffuse neutrino flux both for
the upgoing tracks (orange dashed line) and all-sky showers (red dashed-dotted
line) is shown in Fig. 1. For comparison, the $1~{}\sigma$ bands around the
diffuse neutrino flux from SBGs predicted by [2] from SBGs are also reported.
Predictions shown in the left (right) panel are obtained through a multi-
component fit of the extragalactic gamma-ray background measured by Fermi-LAT
[11] and the 7.5 yr HESE neutrino flux [12] (6 yr shower neutrino flux [13])
measured by IceCube (see [2] for further details).

Figure 1: $90\%$ CL differential sensitivity to a diffuse neutrino flux for
the sample of upgoing tracks (dashed orange) and of all-sky showers (dashed-
dotted red). On the left, the sensitivities are compared to the theoretical
$1\sigma$ band prediction for SBGs neutrino as obtained in [2] through a
multi-component fit of the extragalactic gamma-ray background (EGB) measured
by Fermi-LAT [11], and the 7.5 yr HESE neutrino flux measured by IceCube [12].
On the right, the KM3NeT/ARCA expectations are compared to the blue band which
corresponds to the $1\sigma$ SBG neutrino expectations from [2] obtained by a
multi-component fit of the EGB measured by Fermi-LAT [11] and the 6-yr shower
neutrino flux measured by IceCube [13].

The sensitivities refer to 10 years of data collected by the ARCA detector for
one single flavour of neutrinos. For neutrino energies below $100\,\,\rm TeV$,
the sensitivity is predominantly driven by purely upgoing events $(\theta\leq
80^{\circ})$, where $\theta$ is the reconstructed zenith angle of the tracks
and $\theta=0^{\circ}$ corresponds to a vertical upgoing track. For energies
exceeding $100\,\,\rm TeV$, the sensitivity is dominated by horizontal events
$(80^{\circ}<\theta<100^{\circ})$ due to neutrino absorption in the Earth.
This results in a minimum of the sensitivity around $1\,\rm PeV$. The minimum
of the sensitivity for showers is at $\simeq 100\,\rm TeV$ because the event
containment significantly reduces the amount of observed signal above this
energy. Due to a reduced background rate, the shower sensitivity is better
than the one for tracks for $\rm E_{\nu}\lesssim 100\,\rm TeV$. For $\rm
E_{\nu}\gtrsim 100\,\rm TeV$, the sensitivity is dominated by the tracks,
because of a larger effective volume for track-like events as the energy
increases. After 10 years of operation, ARCA will probe the neutrino emission
from SBGs, either confirming or constraining their contribution to the diffuse
neutrino flux around $\sim 100\,\rm TeV$. The sensitivities shown in Fig. 1
suggest that ARCA will also improve the characterisation of the diffuse
neutrino flux provided by IceCube [12, 13], thus providing valuable
information regarding other astrophysical sources contributing to the diffuse
neutrino flux such as gamma-ray opaque sources [14].

##  5 Point-like SBG Analysis

ARCA is expected to have an excellent angular resolution ($\lesssim
0.2^{\circ}$ for tracks and $\rm E_{\nu}\geq 10\,\rm TeV$ see [7], and
therefore an enhanced sensitivity for point-like and extended sources. The
likelihood function takes into account both the reconstructed energy $(\rm
E_{\rm rec})$ and the angular distance between the nominal position of the
source and the reconstructed direction of the event within an angle $\alpha$.
The range $[10,10^{8}]\,~{}\rm GeV$ and $[0,5^{\circ}]$ $([0,15^{\circ}])$ for
tracks (showers) are respectively considered for the reconstructed energy and
$\alpha$. In the following, the evaluation of the differential limits is
performed considering the position of three local SBGs in order to test the
ability of ARCA to trace their star-forming activity through neutrino
emission. Specifically, the Small Magellanic Cloud and the Circinus Galaxy are
studied, because their expected neutrino fluxes should be high enough to match
ARCA sensitivity after $\sim 6$ years of operation [3]. The case of NGC 1068,
whose neutrino emission has been measured by IceCube at $4.2\sigma$
significance with a neutrino flux normalisation
$\phi_{\nu_{\mu}+\bar{\nu}_{\mu}}=5.0\times 10^{-11}\,\rm TeV^{-1}\,\rm
cm^{-2}\,\rm s^{-1}$ at $\rm E_{\nu}=1\,\rm TeV$ [4], is also considered.

###  5.1 Small Magellanic Cloud

The Small Magellanic Cloud is a local star-forming galaxy at a distance of
$\sim$ $60\,\rm{kpc}$ [15], characterised by a stable and diffuse gamma-ray
emission, driven by its star-forming activity [16]. Its expected neutrino
emission has been estimated using a one-zone model tuned to its gamma-ray
spectrum by [3]. The capability of ARCA to test this model is evaluated by
simulating the SMC as an extended source with a radius $r=0.5^{\circ}$,
consistent with its angular extension measured by the Fermi-LAT Collaboration
[16].

Figure 2: Left: $90\%$ differential sensitivity (orange dashed line) and the
$5\sigma$ differential discovery flux (red dashed-dotted line) of ARCA after
10 years of operation for one neutrino flavour compared with the $1\sigma$ SBG
neutrino model (blue band) from [3]. The energy-integrated sensitivity
(tracks+showers) for the best-fit spectrum of the source (considering 10 years
of operation) is also reported. Right: energy-integrated MRF (best fit flux)
as a function of time for different event samples. The blue line refers to
$\nu_{\mu}$ CC events, orange to the all-flavour tracks, and green to
tracks+showers.

The $90\%$ CL differential sensitivity and the $5\sigma$ differential
discovery flux for the SMC, considering 10 years of ARCA operation, are shown
in the left panel of Fig. 2 and compared with the $1\sigma$ neutrino
expectations from [3]. The track+shower energy-integrated sensitivity is also
reported for the best-fit spectrum of the source which can be described by a
power-law with an exponential cutoff $\sim\rm E^{-2.1}\cdot\rm e^{(-\rm
E/500\,\rm TeV)}$. The cutoff at $500\,\rm TeV$ comes from the assumption of
maximal proton energy $\simeq 10\,\rm PeV$. In the right panel of Fig. 2, The
energy-integrated MRF is shown as a function of time. Three different event
samples are considered: $\nu_{\mu}$ in CC, all-flavour tracks ($\nu_{\mu}$ CC
interactions and $\nu_{\tau}$ in CC interactions where the $\tau$ decays into
a muon) and finally all-flavour tracks +showers. The horizontal threshold line
$(\rm MRF=1)$ is superimposed for comparison. Although the detector is not
sensitive to discriminate the flux in single true energy signal bins, the
integrated spectrum can be constrained after $\sim 8\,\rm years$, combining
the information from the track-like and shower-like sample. The samples
provide different results due to a diverse amount of events (signal and
background) contained in each sample. Therefore, this will test the model
proposed by [3]. It is important to stress that this source, having a low star
formation rate, might have a sizeable contamination from leptonic point-like
sources, such as pulsar wind nebulae, in its gamma-ray spectrum, leading to a
smaller neutrino spectrum than the one provided in [3]. This will allow for a
crucial test of the neutrino content of the source. If the neutrino emission
is limited in a smaller region than the entire galaxy, the sensitivity
improves, as discussed in A.

###  5.2 NGC 1068

NGC 1068 is a source situated at $\sim$10-14 Mpc from the Milky Way [16]. It
is classified as a Starburst and as a type II Seyfert galaxy, characterised by
a star formation rate of $\sim$ $23\,\rm M_{\odot}\,\rm yr^{-1}$ and by the
presence of a starburst ring [17]. Its Seyfert activity is located mainly
inside its active nucleus [16, 18], and its hot corona [16, 17, 19, 20, 21].
The ALMA telescope [17, 22, 23] has also detected a powerful radio jet and
four parsec-scale blobs in the jet head, which may represent sites for cosmic
ray acceleration. Furthermore, a hint of an ultra-fast outflow in the core of
the source has been reported by [24] analysing absorption lines in the X-ray
band. Recently, the IceCube Collaboration has reported an excess of
$79^{+22}_{-20}$ astrophysical neutrinos from the direction of NGC 1068, with
an overall significance of $4.2\sigma$ above the background-only hypothesis
[4]. The measured neutrino flux normalisation is at least a factor 10 above
the measured gamma-ray flux [16, 25], rather than the typical factor of about
2 expected from hadronic emission in transparent sources.
Both AGN and SBG activities have been used to explain the origin of such a
high neutrino flux. However, the SBG-related emission produces a neutrino flux
approximately two orders of magnitude below the IceCube measurements [3, 26].
On the other hand, AGN-related emissions can produce a greater contribution to
observed neutrino flux [27, 28, 29]. At the moment, the favoured scenario
seems a considerable neutrino emission related to the magnetised hot corona
activity [20].

The time-integrated expectations for ARCA are shown in Fig. 3.

Figure 3: Left: $90\%$ CL differential sensitivity (orange dashed line) and
$5\sigma$ differential discovery flux (red dashed-dotted line) and the energy-
integrated $5\sigma$ discovery for an $E^{-3.2}$ spectrum using the
tracks+showers sample (dark red line). The expectations are compared with the
IceCube-measured $2\sigma$ region for the spectrum [4] (blue band) and the
theoretical neutrino predictions from SBG activity according to [3] (grey
band). Right: Energy-integrated $5\sigma$ discovery flux and $90\%$ CL
sensitivity as functions of the operation time based on the combined
track+shower sample. The bands correspond to the the uncertainty of the flux
measured by [4]. After $\sim 3\,\rm years$, KM3NeT/ARCA is expected to confirm
the spectrum measured by IceCube.

In the left panel of Fig. 3, the $90\%$ CL differential sensitivity and the
$5\sigma$ discovery flux for $10\,\rm years$ of data-taking for ARCA are
compared with the IceCube $2\sigma$ region [4] (blue band). The obtained
differential limits show that ARCA can probe a neutrino flux of
$E^{2}\phi_{\nu_{\mu}+\bar{\nu}_{\mu}}\sim 5\cdot 10^{-8}\,\rm GeV\,\rm
cm^{-2}\,\rm s^{-1}$ and of $E^{2}\phi_{\nu_{\mu}+\bar{\nu}_{\mu}}\sim
10^{-9}\,\rm GeV\,\rm cm^{-2}\,\rm s^{-1}$ at $1\,\ \rm TeV$ and $100\,\rm
TeV$, respectively. The energy-integrated 5$\sigma$ discovery for a $E^{-3.2}$
spectrum and the expected SBG neutrino spectral energy distribution evaluated
according to [3] are also shown for comparison.

In the right panel of Fig. 3, the energy-integrated discovery flux and MRF are
reported as functions of observation time and compared with the discovery
threshold corresponding to $\rm MDP=1$. The bands refers to the uncertainty of
the flux measured by [4].

The ARCA detector is expected to confirm the flux reported as a best fit by
IceCube after only $\sim 3\,\rm years$ of data taking and to better
disentangling the emission from AGN-related components from the SBG-related
one.

###  5.3 Circinus Galaxy

Circinus is a SBG located $\sim$ $4\,\rm Mpc$ from the Milky Way [15] at a
declination $\delta=-65.2^{\circ}$ [30], where the ARCA detector is expected
to have full visibility [9]. Circinus is classified as a Seyfert II galaxy
[31], showing both AGN activity and a hot corona [20, 32]. The neutrino flux
emission predicted by the hot corona model can reach a flux level of
$E^{2}\phi_{\nu_{\mu}+\bar{\nu}_{\mu}}\simeq 10^{-8}\,\rm GeV\,\rm
cm^{-2}\,\rm s^{-1}$ [20] above $\rm E_{\nu}\sim 1\,\rm TeV$.

Figure 4: Left: $90\%$ CL track differential sensitivity (dashed orange line),
$5\sigma$ track differential discovery flux (dashed-dotted red line) for 10
years of data taking with the ARCA detector, compared with the $1\sigma$ SBG
neutrino flux predictions from [3] (blue band). Also shown is the energy-
integrated $90\%$ CL sensitivity for $E^{-2.3}\cdot\rm e^{-\rm E/500\,\rm
TeV}$, corresponding to the best-fit neutrino spectrum from SBG activity.
Right: The MRF as a function of observation time for the best-fit spectrum of
the source (dashed-dotted blue line, all-flavour tracks), for the highest
possible flux $(1\sigma)$ (dashed orange line, all-flavour tracks) and for the
highest possible flux considering tracks and showers (green line).

In the left panel of Fig. 4, the $90\%$ CL differential sensitivity and the
$5\sigma$ differential discovery flux are compared with the $1\sigma$ neutrino
expectations from SBG activity [3]. The differential limits indicate that a
flux of the order of $E^{2}\phi_{\nu_{\mu}+\bar{\nu}_{\mu}}\simeq 10^{-9}\,\rm
GeV\rm cm^{-2}\,\rm s^{-1}$ can be probed in the energy range $30-100\,\rm
TeV$. Therefore, a neutrino flux connected to the AGN activity can be searched
for with ARCA, which may lead to the discovery of a hot corona emission like
in NGC 1068. The energy-integrated sensitivity for the best-fit SBG flux $(\rm
E^{-2.3}\cdot e^{(-\rm E/500\,\rm TeV)})$ is also reported. In the right panel
of Fig.4, the MRF is presented as a function of observation time for different
cases: the best-fit flux of the source analysed with all-flavours tracks and
the highest possible flux $(1\sigma)$ analysed with all-flavour tracks and
all-flavour tracks + showers. The obtained sensitivity shows that the latter
can be constrained after $\sim 20\,$ years of observations.

##  6 Impact of Systematic Uncertainties

In this section, a brief discussion on the systematic uncertainties associated
with the ARCA sensitivity is presented. The most significant factors impacting
the sensitivity are a $25\%$ uncertainty in the conventional atmospheric
neutrino flux and a $10\%$ uncertainty in the effective area [1]. The
resulting uncertainty on the sensitivity from these effects can be
conservatively estimated by using the fact that, in background-dominated data,
$\rm MRF\propto\sqrt{\rm n_{back}}/\rm n_{s}$, where $\rm n_{back}$ is the
expected number of background events and $\rm n_{s}$ is the expected number of
signal events provided by the input spectrum [10]. Since both the numbers of
events (signal and background) are proportional to the effective area, a
variation of 10% leads to a $4.6\%$ uncertainty on the sensitivity.
Furthermore, conservatively assuming that the background sample is dominated
by atmospheric neutrinos, an uncertainty of $25\%$ on the background rate
leads to a $12.5\%$ uncertainty on the sensitivity. Regarding the uncertainty
of the discovery flux, Ref. [9] has shown that the $25\%$ uncertainty on the
atmospheric neutrino flux leads to a $+15\%$ and $-5\%$ uncertainty.
Therefore, it is expected that for the SMC, the time required to achieve a 90%
CL constraint varies within $8.0^{+2.2}_{-1.7},\rm years$, whereas for the
highest possible flux from the Circinus galaxy, it is $20\pm 5,\rm years$. In
contrast, the discovery of the $E^{-3.2}$ spectrum observed by IceCube in the
direction of NGC 1068 yields an estimated time of $3.0^{+1.0}_{-0.3},\rm yr$.

##  7 Final Remarks and Conclusions

In this paper, the expected differential sensitivity for the ARCA detector is
presented for the first time for a diffuse neutrino flux and individual
neutrino sources. A binned maximum likelihood formalism is applied to evaluate
the $90\%$ CL differential sensitivity and the $5\sigma$ differential
discovery flux, by binning the signal flux in true energy bins of half-decade
widths. The differential limits are shown to be mostly signal model-
independent and complementary to the energy-integrated limits and provide
crucial information regarding the energy ranges where the detector is more
sensitive to neutrino signals. They are evaluated using both upgoing track-
like and all-sky contained shower-like events.
The results show that ARCA will probe the shape of the diffuse flux for
$E\lesssim 100\,\rm TeV$, constraining the role of sources such as SBGs and
complementing the IceCube’s observations of the neutrino sky.
The likelihood formalism is also applied to local SBG sources, following the
flux predictions provided in [3]. After 8 years of data taking, the Small
Magellanic Cloud can be observed as a $90\%$ CL excess in the energy-
integrated analysis, proving that SBGs are guaranteed neutrino emitters. ARCA
expectations for NGC 1068 have been evaluated indicating that ARCA will need
only$\sim$3 years for discovering this source with $5\sigma$ significance.
This will better disentangle the AGN-related contributions from the SBG-
related one, strengthening IceCube observations. Finally, the differential
limits for the Circinus Galaxy are evaluated, showing that the detector can
probe its potential AGN-related emission activity additional to the expected
SBG neutrino flux.

##  8 Acknowledgements

The authors gratefully acknowledge the contribution from Damiano F. G.
Fiorillo and Marco Chianese for providing support regarding SBG fluxes and for
fruitful discussions.

The authors acknowledge the financial support of the funding agencies:

Czech Science Foundation (GAČR 24-12702S);

Agence Nationale de la Recherche (contract ANR-15-CE31-0020), Centre National
de la Recherche Scientifique (CNRS), Commission Européenne (FEDER fund and
Marie Curie Program), LabEx UnivEarthS (ANR-10-LABX-0023 and
ANR-18-IDEX-0001), Paris Île-de-France Region, France;

Shota Rustaveli National Science Foundation of Georgia (SRNSFG, FR-22-13708),
Georgia;

The General Secretariat of Research and Innovation (GSRI), Greece;

Istituto Nazionale di Fisica Nucleare (INFN) and Ministero dell’Università e
della Ricerca (MUR), through PRIN 2022 program (Grant PANTHEON 2022E2J4RK,
Next Generation EU) and PON R&I program (Avviso n. 424 del 28 febbraio 2018,
Progetto PACK-PIR01 00021), Italy; A. De Benedittis, R. Del Burgo, W. Idrissi
Ibnsalih, A. Nayerhoda, G. Papalashvili, I. C. Rea, S. Santanastaso, A.
Simonelli have been supported by the Italian Ministero dell’Università e della
Ricerca (MUR), Progetto CIR01 00021 (Avviso n. 2595 del 24 dicembre 2019);

Ministry of Higher Education, Scientific Research and Innovation, Morocco, and
the Arab Fund for Economic and Social Development, Kuwait;

Nederlandse organisatie voor Wetenschappelijk Onderzoek (NWO), the
Netherlands;

The National Science Centre, Poland (2021/41/N/ST2/01177); The grant
“AstroCeNT: Particle Astrophysics Science and Technology Centre”, carried out
within the International Research Agendas programme of the Foundation for
Polish Science financed by the European Union under the European Regional
Development Fund;

National Authority for Scientific Research (ANCS), Romania;

Slovak Research and Development Agency under Contract No. APVV-22-0413;
Ministry of Education, Research, Development and Youth of the Slovak Republic;

MCIN for PID2021-124591NB-C41, -C42, -C43 funded by MCIN/AEI/
10.13039/501100011033 and, as appropriate, by “ERDF A way of making Europe”,
by the “European Union” or by the “European Union NextGenerationEU/PRTR”,
Programa de Planes Complementarios I+D+I (refs. ASFAE/2022/023,
ASFAE/2022/014), Programa Prometeo (PROMETEO/2020/019) and GenT (refs.
CIDEGENT/2018/034, /2019/043, /2020/049. /2021/23) of the Generalitat
Valenciana, Junta de Andalucía (ref. SOMM17/6104/UGR, P18-FR-5057), EU: MSC
program (ref. 101025085), Programa María Zambrano (Spanish Ministry of
Universities, funded by the European Union, NextGenerationEU), Spain; The
European Union’s Horizon 2020 Research and Innovation Programme (ChETEC-INFRA
- Project no. 101008324).

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##  Appendix A Impact of Extension to the Differential Sensitivity

The differential sensitivity at the declination of the Small Magellanic Cloud
after 10 years of operation of the ARCA detector (for the tracks) is shown in
Fig. 5 considering a point-like source and an extended source with
$r=0.5^{\circ}$.

Figure 5: Comparison between the $90\%$ CL differential sensitivity for a
point-like source (orange dashed line) and for an extended source with an
extension of $r=0.5^{\circ}$ (blue dashed-dotted line), considering 10 years
of data-taking at the declination of the SMC.

The differential sensitivity of the extended source worsens by $\sim$$30\%$
with respect to the point-like source case. Therefore, in case the neutrino
emission for SBGs is concentrated within their nuclei, the sensitivity is
expected to be close to the one expected for point-like sources and the time
required for a $90\%$ CL excess might reduce with the one presented in the
main text.

##  Appendix B Sensitivity Dependence on Declination

The comparison between the differential sensitivities after 10 years of
operation of ARCA is shown in Fig. 6 for three declinations:
$\delta=-73^{\circ}$, $\delta=-65.3^{\circ}$ and finally
$\delta=-0.01^{\circ}$. These declinations correspond to the positions of the
SMC, Circinus Galaxy, and NGC 1068, respectively.

Figure 6: Comparison between point-like $90\%$ CL differential sensitivity for
several declinations after 10 years of operation of the ARCA detector. The
blue dashed-dotted line refers to $\delta=-73^{\circ}$, the orange dashed line
to $\delta=-65^{\circ}$ and the green line to $\delta=-0.01^{\circ}$.

The sensitivities for $\delta=-73^{\circ}$ and $\delta=-65^{\circ}$ are very
close as the declination bands are also near. At high energy the sensitivity
deteriorates faster for $\delta=-73^{\circ}$ due to absorption from the Earth.
The sensitivity at $\delta=-0.01^{\circ}$ is better at high energies due to a
higher signal rate, but it is worse than the others between $10-300\,\rm TeV$
due to a higher background rate. In fact, $\delta=-0.01^{\circ}$, the sources
are only partially below the horizon (see for instance [9] for the ARCA
detector visibility as a function of declination).


have some $j$ such that the first, second and third clauses of
$R_{0}(p,\hat{p},q,j)$ hold, so that $R_{0}(p,\hat{p},q,j)$ holds. But then
$a+1$ appears in the output where $a=\langle
p_{j},\overline{2^{-j+1}}\rangle$, in particular
$x\in\alpha(a)\subseteq(\delta_{\Sigma^{0}_{1}(X)}\circ F)(p,\hat{p},q)$.

Thus we have shown, given $\hat{p}\in(\delta^{\prime})^{-1}\check{Z}$,
$q\in\delta_{\Sigma^{0}_{1}(X)}^{-1}\\{U\\}$, $p\in\delta_{X}^{-1}\\{x\\}$
where $x\in(h\circ r)^{-1}U$, that $(h\circ
r)^{-1}U\subseteq\bigcup\\{(\delta_{\Sigma^{0}_{1}(X)}\circ
F)(p,\hat{p},q)\mid p\in\delta_{X}^{-1}(h\circ
r)^{-1}\delta_{\Sigma^{0}_{1}(K)}(q)\\}$. This completes the proof.

##  6 Zero dimensional subsets

In [Ken15, §4], the present author introduced four ‘natural’ operations
$N,M,B,S$ concerning a general represented class
$\mathcal{Y}\subseteq\\{A\subseteq X\mid\dim A\leq 0\\}$, finding that $N$ is
computable only if $M$ is computable, $M$ computable only if $B$ computable,
and $B$ computable only if $S$ is computable. Here, we specialize to
$\mathcal{Y}\subseteq\\{A\in\Pi^{0}_{1}(X)\mid\dim A\leq 0\\}$ with
$\delta_{\mathcal{Y}}\leq\delta_{\Pi^{0}_{1}}|^{\mathcal{Y}}$, and find the
respective conditions of computability of the operations $N,M,B,S$ are in fact
equivalent in this restricted case. More specifically, we consider also
$N_{0}:\subseteq\Pi^{0}_{1}(X)^{2}\times\mathcal{Y}\rightrightarrows\Sigma^{0}_{1}(X)^{2}$
defined by

$N_{0}(A,B,Y):=\\{(U,V)\mid A\cap Y\subseteq U\wedge B\cap Y\subseteq V\wedge
Y\cap U\cap V=\emptyset\wedge Y\subseteq U\cup V\\}$

($\operatorname{dom}N_{0}=\\{(A,B,Y)\mid A\cap B\cap Y=\emptyset\\}$) and show
$N\equiv_{\textrm{W}}N_{0}$ and $N_{0}\leq_{\textrm{sW}}S$. Here if
$f:\subseteq W\rightrightarrows X$ and $g:\subseteq Y\rightrightarrows Z$, we
have _Weihrauch reducibility_ and _strong Weihrauch reducibility_ respectively
defined by

$\displaystyle f\leq_{\textrm{W}}g$ $\displaystyle:\iff(\exists\text{
computable
}H,K:\subseteq\mathbb{N}^{\mathbb{N}}\to\mathbb{N}^{\mathbb{N}})\left(\forall\text{
realizer }G\text{ of }g\right)$
$\displaystyle\left(F:=H\circ\langle\operatorname{id}_{\mathbb{N}^{\mathbb{N}}},G\circ
K\rangle\text{ a realizer of }f\right),$ $\displaystyle f\leq_{\textrm{sW}}g$
$\displaystyle:\iff(\exists\text{ computable
}H,K:\subseteq\mathbb{N}^{\mathbb{N}}\to\mathbb{N}^{\mathbb{N}})\left(\forall\text{
realizer }G\text{ of }g\right)$ $\displaystyle\left(F:=H\circ G\circ K\text{ a
realizer of }f\right).$

(For more on these notions, the reader can refer e.g. to [BGP18]). Recall
$N:\subseteq\Pi^{0}_{1}(X)^{2}\times\mathcal{Y}\rightrightarrows\Sigma^{0}_{1}(X)^{2}$
is defined by $\operatorname{dom}N=\\{(A,B,Y)\mid A\cap B=\emptyset\\}$ and
$N(A,B,Y):=\\{(U,V)\mid A\subseteq U\wedge B\subseteq V\wedge Y\subseteq
U\dot{\cup}V\\}$.

######  Proposition 30.

If $\mathcal{Y}\subseteq\\{A\in\Pi^{0}_{1}(X)\mid\dim A\leq 0\\}$ with
$\delta_{\mathcal{Y}}\leq\delta_{\Pi^{0}_{1}}|^{\mathcal{Y}}$ then
$N\leq_{\textrm{W}}N_{0}$.

######  Proof 6.1.

First, we define computable $H,K:\subseteq\mathbb{B}\to\mathbb{B}$ such that
$F:=H\circ\langle\operatorname{id}_{\mathbb{B}},G\circ K\rangle$ realizes $N$
whenever $G$ is a realizer of $N_{0}$. Namely, let
$K:=\operatorname{id}_{\mathbb{B}}$ and if $I,J,L$ are respective computable
realizers of $\cap:\Pi^{0}_{1}(X)^{2}\to\Pi^{0}_{1}(X)$,
$\delta_{\mathcal{Y}}\leq\delta_{\Pi^{0}_{1}(X)}|^{\mathcal{Y}}$ and
$\cup:\Pi^{0}_{1}(X)^{2}\to\Pi^{0}_{1}(X)$, and $P$ a computable realizer of
$t_{4}$, we let $H:\subseteq\mathbb{B}\to\mathbb{B}$ be defined by

$H\langle p,q\rangle:=P\langle L\langle\pi_{1}\pi_{1}p,I\langle
J(\pi_{2}p),\pi_{2}q\rangle\rangle,L\langle\pi_{2}\pi_{1}p,I\langle
J(\pi_{2}p),\pi_{1}q\rangle\rangle\rangle.$

Clearly $H$ is computable, so we verify $F$ realizes $N$. Namely, let
$(A,B,Y)\in\operatorname{dom}N$,
$p\in[\delta_{\Pi^{0}_{1}}^{2},\delta_{\mathcal{Y}}]^{-1}\\{(A,B,Y)\\}$,
$q:=G(p)\in(\delta_{\Sigma^{0}_{1}}^{2})^{-1}N_{0}(A,B,Y)$, so
$(U,V):=\delta_{\Sigma^{0}_{1}}^{2}(q)$ have

$\displaystyle A\cup(Y\setminus V)=(\delta_{\Pi^{0}_{1}}\circ
L)\langle\pi_{1}\pi_{1}p,I\langle
J(\pi_{2}p),\pi_{2}q\rangle\rangle=:C\quad\text{ and }$ $\displaystyle
B\cup(Y\setminus U)=(\delta_{\Pi^{0}_{1}}\circ
L)\langle\pi_{2}\pi_{1}p,I\langle
J(\pi_{2}p),\pi_{1}q\rangle\rangle\rangle=:D,$

in particular $H\langle
p,q\rangle\in(\delta_{\Sigma^{0}_{1}}^{2})^{-1}t_{4}(C,D)$. Denoting
$(U^{\prime},V^{\prime}):=(\delta_{\Sigma^{0}_{1}}^{2}\circ H)\langle
p,q\rangle$ we have $A\subseteq C\subseteq U^{\prime}$, similarly $B\subseteq
V^{\prime}$ and we will see $Y\subseteq U^{\prime}\dot{\cup}V^{\prime}$ (hence
$(U^{\prime},V^{\prime})\in N(A,B,Y)$): namely $Y\subseteq U\cup V$ with
$Y\cap U=Y\setminus V\subseteq C\subseteq U^{\prime}$, similarly $Y\cap
V\subseteq V^{\prime}$ (and already $U^{\prime}\cap V^{\prime}=\emptyset$ by
definition of $t_{4}$).

Conversely, to show $N_{0}\leq_{\textrm{sW}}N$, if $G$ is a realizer of $N$,
$(A,B,Y)\in\operatorname{dom}N_{0}$ and $I$, $J$ are computable realizers of
$\cap:\Pi^{0}_{1}(X)^{2}\to\Pi^{0}_{1}(X)$ and
$\delta_{\mathcal{Y}}\leq\delta_{\Pi^{0}_{1}(X)}|^{\mathcal{Y}}$, then we have
$(A\cap Y)\cap(B\cap Y)=A\cap B\cap Y=\emptyset$ so $(A\cap Y,B\cap
Y,Y)\in\operatorname{dom}N$, and we can consider computing some $(U,V)\in
N(A\cap Y,B\cap Y,Y)$. Then $A\cap Y\subseteq U\wedge B\cap Y\subseteq V\wedge
Y\subseteq U\dot{\cup}V$ implies $U\cap V\cap Y=\emptyset$ and $(U,V)\in
N_{0}(A,B,Y)$. On the other hand, we will compare $N_{0}$ with $S$, defined as
follows:
$S:\subseteq\Sigma^{0}_{1}(X)^{\mathbb{N}}\times\mathcal{Y}\rightrightarrows\Sigma^{0}_{1}(X)^{\mathbb{N}}$,
$\operatorname{dom}S=\\{((U_{i})_{i},Y)\mid\bigcup_{i}U_{i}\supseteq Y\\}$,
$S((V_{i})_{i},Y):=\\{(W_{i})_{i}\mid(\forall i)W_{i}\subseteq
V_{i}\wedge\bigcup_{i}W_{i}\supseteq Y\text{ and $(W_{i})_{i}$ pairwise
disjoint}\\}$.

######  Proposition 31.

$N_{0}\leq_{\textrm{sW}}S$ holds for any represented class
$\mathcal{Y}\subseteq\\{A\subseteq X\mid\dim A\leq 0\\}$.

######  Proof 6.2.

Let
$K(p):=\langle\langle\pi_{2}\pi_{1}p,\pi_{1}\pi_{1}p,0^{\mathit{\omega}},0^{\mathit{\omega}},\dots\rangle,\pi_{2}p\rangle$
and $H\langle q^{(0)},q^{(1)},\dots\rangle:=\langle q^{(0)},q^{(1)}\rangle$;
then any realizer $G$ of $S$ gives rise to a realizer $F:=H\circ G\circ K$ of
$N_{0}$. Namely, if $(A,B,Y)\in\operatorname{dom}N_{0}$,
$p\in[\delta_{\Pi^{0}_{1}}^{2},\delta_{\mathcal{Y}}]^{-1}\\{(A,B,Y)\\}$ and
$q:=\langle q^{(0)},\dots\rangle:=(G\circ K)(p)$ then
$q\in(\delta_{\Sigma^{0}_{1}}^{\mathit{\omega}})^{-1}S(X\setminus B,X\setminus
A,\emptyset,\emptyset,\dots;Y)$. Denoting
$(W_{i})_{i}=\delta_{\Sigma^{0}_{1}}^{\mathit{\omega}}(q)$ by definition of
$S$ we have $W_{0}\cap B=\emptyset=W_{1}\cap A\wedge Y\subseteq
W_{0}\dot{\cup}W_{1}$. But then $A\cap Y\subseteq W_{0}$, $B\cap Y\subseteq
W_{1}$ and $Y\cap W_{0}\cap W_{1}=\emptyset$, so $(W_{0},W_{1})\in
N_{0}(A,B,Y)$, and $(W_{0},W_{1})=(\delta_{\Sigma^{0}_{1}}^{2}\circ F)(p)$.

##  7 Concluding Remarks

The results of this paper, in particular the reductions of Section 6 along
with Theorem 18 and Proposition 22, provide some details on a robust notion of
effective zero-dimensionality for classes $\mathcal{Y}$ of subsets of a
computable metric space $X$, extending the notion of effective zero-
dimensionality for $X$ that was examined in [Ken15, §5]. It would be good to
find precisely the necessary compactness and other assumptions for Proposition
22, and to clarify in terms of Weihrauch reducibility the results of Theorem
18 and Proposition 22, at least for classes of closed subsets.

On the other hand, in Theorem 28 we have followed the proof of the Dugundji
Extension Theorem outlined in [Eng89, Hint to Prob 4.5.20(a)] to compute a
more general form of retraction, seemingly requiring a valued field structure
(and use of convexity in ultrametric spaces, along with the proof technique of
Theorem 18) in the process. It is interesting (though perhaps more
complicated) to ask whether some weaker algebraic conditions would suffice,
e.g. what the ‘general form’ of retractions should be in zero-dimensional
topological groups? Here note, regarding valued fields, we have stated Theorem
28 partly for a zero-dimensional computable metric space $X$ and partly for a
$p$-adic field $K=\Omega_{p}$, with the following well-known embedding theorem
in mind:

######  Theorem 32.

(see [Sch84, §A.10, Corollary]) Any separable ultrametric space $X$ has an
isometric embedding into some separable nonarchimedean valued field $K$.

An effective version of this result may be of interest, in part to simplify
the assumptions of Theorem 28, and in any case the notions appearing in
[Sch84, §18–21] (on ultrametric spaces; see also the references of [KKS13])
deserve further attention from a computable analysis viewpoint. As regards
$p$-adic fields or slightly different number-theoretic settings, there are
very many concrete tools (and topics of analysis) in the remainder of [Sch84]
whose relation to computable analysis (and possibly effective zero-
dimensionality) may be worth investigating, though our knowledge is limited.
To mention two sections close to the topics already discussed, [Sch84, §26,
§76] respectively concern locally constant ($K$-valued) functions &
differentiability, and extension of functions (including isometries).

Also, [Eng89, Prob 6.3.2(f)] supplies conditions on a zero-dimensional
computable metric space $X$ under which the topology is induced by a linear
order $(<)\subseteq X^{2}$.

Finally, computability of $Q^{\prime}$ in the proof of Proposition 11 has
shown how a lower semicontinuous function $f:X\setminus A\to(0,1]$ satisfying
certain nonuniform upper bounds (dependent on $\epsilon>0$) computably gives
rise to a Dugundji system for $A\in\mathcal{A}(X)\setminus\\{\emptyset,X\\}$
(with coefficient $1+\epsilon$), without involving zero-dimensionality. Here
it is of interest to see how such an $f$ may be chosen to be lower
semicomputable without necessarily being continuous. To provide context, some
results on lower semicontinuous functions and references to applications
appear in [WZ00]; here we leave unanswered the question of applications of the
general Dugundji system construction, and instead remark on two related
questions around lower semicontinuity:

  1. 1.

If $(X,\mathcal{T}_{X},\alpha)$ is an effective topological space, each
characteristic function $\chi_{U}:X\to\mathbb{R}$ ($U\in\Sigma^{0}_{1}(X)$) is
$(\delta_{X},\rho_{<})$-continuous, and $u:\mathbb{N}\times
X\to\mathbb{R},(i,x)\mapsto\chi_{\alpha(i)}(x)$ is
$(\operatorname{id}_{\mathbb{N}},\delta_{X};\rho_{<})$-computable. More
generally,

$s:\subseteq\mathbb{N}\times\mathbb{B}\times\mathbb{B}^{\mathbb{N}}\to\mathit{C}_{\Sigma^{0}_{1}(X)}(X,\mathbb{R}_{<}),(N,p;p^{(0)},\dots)\mapsto\sum_{i<N}\rho_{<}(p^{(i)}).\chi_{\alpha^{\prime}(p_{i})}|_{\delta_{\Sigma^{0}_{1}(X)}(p)}$

(with $\operatorname{dom}s=\\{(N,p;p^{(0)},\dots)\mid(\forall
i<N)p^{(i)}\in\rho_{<}^{-1}[0,\infty)\\}$) is
$(\operatorname{id}_{\mathbb{N}},\operatorname{id}_{\mathbb{B}},\operatorname{id}_{\mathbb{B}}^{\mathit{\omega}};[\delta_{X}\to\rho_{<}]_{\Sigma^{0}_{1}(X)})$-computable.

One might consider those
$f\in\mathit{C}_{\Sigma^{0}_{1}(X)}(X,\mathbb{R}_{<})$ which lie in the range
of $s$; in the case that $\operatorname{im}\alpha\subseteq\Delta^{0}_{1}(X)$
we know such functions are locally constant; for $K$-valued functions (in
particular, for rational-valued functions when $X$ is the set of $p$-adic
integers in $K=\Omega_{p}$) that would also imply $f^{\prime}=0$ under
suitable assumptions on $X$ and $\operatorname{dom}f$; see [Sch84, §26].

One can note further that a certain sequence of locally constant functions
(from the $p$-adic integers to $K$) is used to define a basis of the
$\mathit{C}^{1}$ functions with zero derivative (with the same domain and
codomain) in [Sch84, Thm 68.1]; we leave this to one side here and just
discuss the locally constant functions. Under one weak definition of monotone
sequences, any continuous function $f:X\to K$ is the uniform limit of a
monotone sequence of locally constant $f_{n}:X\to\mathbb{Q}$
($n\in\mathbb{N}$), cf. [Sch84, Ex 87.H(i)]. It is of interest to examine
whether this or another sense of monotone sequence (of locally constant
functions) gives rise to representations of $\mathit{C}(X,K)$ analogous to the
restriction $[\delta_{X}\to\rho_{<}]|^{\mathit{C}(X,\mathbb{R})}$ in the real
case.

  2. 2.

If $X$ is a complete computable metric space, $\mathcal{X}$ the Borel
$\sigma$-algebra of $X$ and $\mu:\mathcal{X}\to[0,1]$ is a probability
measure, recall $\mu$ is a _computable probability measure on $X$_ if
$\mathbb{N}^{*}\to\mathbb{R}_{<},w\mapsto\mu\left(\bigcup_{i<\lvert
w\rvert}\alpha(w_{i})\right)$ is computable. Note this is equivalent to
$g_{\mu}:=\mu|_{\Sigma^{0}_{1}(X)}:\Sigma^{0}_{1}(X)\to\mathbb{R}_{<}$ being
computable, e.g. [HR09, Thm 4.2.1]. More generally, we can introduce a
representation of the set $\mathcal{M}(X)$ of finite measures
$\mu:\mathcal{X}\to[0,\infty)$ by

$p\in\delta_{\mathcal{M}(X)}^{-1}\\{\mu\\}\iff
p\in[\delta_{\Sigma^{0}_{1}(X)}\to\rho_{<}]^{-1}\\{g_{\mu}\\}.$

If the construction of a Dugundji system in (the proof of) Proposition 11 can
be adequately generalised to nonmetrizable spaces like $Z=\Sigma^{0}_{1}(X)$
(with the topology from Remark 13), then a restriction of $g_{\mu}$ (for a
probability measure $\mu\in\mathcal{M}(X)$) to a specified open subspace of
$Z$ may provide an example of a lower semicomputable function $f:\subseteq
Z\to(0,1]$ with open domain. (Note if $Y\in\Sigma^{0}_{1}(X)$ is a proper
nonempty open subset then $A:=\Sigma^{0}_{1}(Y)$ is closed and nonopen in
$Z$).

Of course, in the setting of $p$-adic fields there are several analogues of
measure and integration (both $K$-valued and real-valued) appearing in
[Sch84], which may also be worth considering in this connection.

## Acknowledgement

The author is very grateful for the useful comments of the anonymous referees,
which have helped to improve the paper, inclusive of correcting some errors.

## References

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# Separately regular and separately Nash functions

Juliusz Banecki

###### Abstract

We strengthen certain known results saying that separately regular functions
are rational and separately Nash functions are semialgebraic. The approach
presented here unifies and highlights the similarities between the two
problems.

Given three sets $X,Y,Z$ a function $f:X\times Y\rightarrow Z$ and points
$x_{0}\in X,y_{0}\in Y$ we denote the functions $x\mapsto f(x,y_{0})$ and
$y\mapsto f(x_{0},y)$ by $f(\cdot,y_{0})$ and $f(x_{0},\cdot)$ respectively.

Let $\mathbb{K}$ be an infinite field. By an affine variety over $\mathbb{K}$
we mean a Zariski closed subset $X$ of $\mathbb{K}^{n}$ for some $n$. A
function $f:X\rightarrow\mathbb{K}$ is polynomial if it is the restriction of
some polynomial function on $\mathbb{K}^{n}$. The algebra of polynomial
functions on $X$ is denoted by $\mathcal{P}(X)$. The function $f$ is said to
be regular if for every point $x\in X$ there exist two polynomial functions
$P,Q\in\mathcal{P}(X)$ such that $Q(x)\neq 0$ and $Qf=P$ holds on the entire
variety $X$. The algebra of regular functions on $X$ is denoted by
$\mathcal{R}(X)$.

Following [4] an irreducible algebraic affine variety $X$ is said to be
_algebraically of the second category_ , if whenever it is covered by a
countable family of subsets

$X=\bigcup_{i=1}^{\infty}X_{i}$

at least one of the sets $X_{i}$ is Zariski dense in $X$.

######  Example 0.1.

According to [4, Theorem 9.3] if $\mathbb{K}$ is an uncountable algebraically
closed field then every irreducible affine algebraic variety $X$ over
$\mathbb{K}$ is algebraically of the second category. Also thanks to [2, Lemma
5] if $\mathbb{K}$ is any uncountable field then $X=\mathbb{K}^{n}$ is
algebraically of the second category for every $n\geq 0$.

This first main result of the paper is the following strengthening of a result
contained in [2]:

######  Theorem A.

Let $\mathbb{K}$ be a field and let $X,Y$ be two affine irreducible varieties
over $\mathbb{K}$. Suppose that $Y$ is algebraically of the second category.
Let $A\subset X$ be a Zariski dense set. Suppose that a mapping $f:X\times
Y\rightarrow\mathbb{K}$ satisfies the following conditions:

  1. 1.

$f(\cdot,y)$ is regular for $y\in Y$

  2. 2.

$f(x,\cdot)$ is regular for $x\in A$.

Then $f$ is rational, i.e. there exist two polynomials
$P,Q\in\mathcal{P}(X\times Y)$ with $Q$ non-zero such that $Qf=P$ on $X\times
Y$.

The proof is based on ideas contained in [2].

Let now $R$ be a real closed field. A subset $X\subset R^{n}$ for some $n$ is
said to be semialgebraic if it is given by some sign conditions on a finite
family of polynomials in $\mathcal{P}(R^{n})$. Let $X\subset R^{n}$ be an open
semialgebraic set. A function $f:X\rightarrow R$ is said to be Nash if it is
infinitely differentiable and semialgebraic. For an introductory treatment of
the topic see [1, Section 8].

We prove the following version of Theorem A for Nash mappings:

######  Theorem B.

Let $R$ be an uncountable real closed field. Let $X$ and $Y$ be two
semialgebraic open subsets of $R^{n_{X}}$ and $R^{n_{Y}}$ respectively, for
some integers $n_{X},n_{Y}\geq 0$. Assume that $X$ is semialgebraically
connected. Let $A\subset X$ be a set dense in the Zariski topology on
$R^{n_{X}}$. Suppose that a mapping $f:X\times Y\rightarrow R$ satisfies the
following conditions:

  1. 1.

$f(\cdot,y)$ is Nash for any $y\in Y$

  2. 2.

$f(x,\cdot)$ is Nash for any $x\in A$

Then $f$ is semialgebraic.

This is a generalisation of some results contained in [3]. The proof is based
on ideas coming from [2], however in this case much more work needs to be done
than in the proof of Theorem A.

##  1 Proofs

###  1.1 The main lemma

We need to state a lemma which will play a role both in the proof of Theorem A
and Theorem B. It seems like the lemma may be useful in proving other results
similar to the ones in the paper, and for this reason it is stated in a rather
general form.

We need to introduce some notation. Let $\mathbb{K}$ be a field and let $S$ be
a set. We denote by $\mathbb{K}^{S}$ the vector space of all $\mathbb{K}$
valued functions on $S$. Let $V\subset\mathbb{K}^{S}$ be some vector subspace.
We denote by $T^{k}V$ the $k$th tensor power of $V$, which naturally embeds in
$K^{S^{k}}$ with the embedding defined as the linear extension of

$(Q_{1}\otimes\dots\otimes Q_{k})(x_{1},\dots,x_{k}):=Q_{1}(x_{1})\dots
Q_{k}(x_{k})$

Suppose further that $V$ admits a countable basis $(e_{i})_{i\in\mathbb{N}}\in
V$. Let $(e_{i}^{\ast})_{i\in\mathbb{N}}\in V$ denote the duals. For an
element $Q\in V\backslash\\{0\\}$ define $d(Q)=\max\\{i:e_{i}^{*}(Q)\neq
0\\}$. For a subset $F\subset S$ define $c(F):=\min\\{d(Q):Q\in
V\backslash\\{0\\},\;Q|_{F}\equiv 0\\}$ (take $c(F)=\infty$ if the set is
empty).

######  Lemma 1.1.

Let $S$ be a set and $V\subset\mathbb{K}^{S}$ a vector space with countable
basis $(e_{i})_{i\in\mathbb{N}}$ as above. Let $n$ be a natural number. There
exist $n$ elements $(P_{i})_{i=1,\dots,n}$ of $T^{n-1}V$ such that for any
subset $F\subset S$ satisfying $c(F)=n$ and any $n-1$ points
$x_{1},\dots,x_{n-1}\in F$ if we define

$P:=\sum_{i=1}^{n}P_{i}(x_{1},\dots,x_{n-1})e_{i}\in V$

then $P|_{F}\equiv 0$. Moreover, for a given set $F$ the points
$x_{1},\dots,x_{n-1}$ can be chosen in such a way that $P\neq 0$.

###### Proof.

Fix the set $F\subset S$. Consider the subspace of $\mathbb{K}^{F}$ spanned by
the functions $e_{1}|_{F},\dots,e_{n}|_{F}$. Its dimension is precisely $n-1$
as the only non-trivial relation between the vectors is given by the
coefficients of an element $Q\in V\backslash\\{0\\}$ satisfying
$d(Q)=n,Q|_{F}\equiv 0$.

It is a simple fact from linear algebra that we can find $n-1$ distinct points
$x_{1},\dots,x_{n-1}\in F$ such that

$e_{1}|_{\\{x_{1},\dots,x_{n-1}\\}},\dots,e_{n}|_{\\{x_{1},\dots,x_{n-1}\\}}$

span a subspace of $\mathbb{K}^{\\{x_{1},\dots,x_{n-1}\\}}$ of dimension
$n-1$. This precisely means that the matrix

$M:=\begin{bmatrix}e_{1}(x_{1})&\dots&e_{n}(x_{1})\\\ \vdots&\ddots&\vdots\\\
e_{1}(x_{n-1})&\dots&e_{n}(x_{n-1})\end{bmatrix}$

is of rank $n-1$.

As already said, up to scalar multiplication the unique non-zero vector in the
kernel of $M$ is given by $[e_{i}^{\ast}(Q)]_{i=1,\dots,n}$. On the other hand
if by $\delta_{i}$ we denote the determinant of $M$ with $i$-th column removed
times $(-1)^{i}$ then the vector $[\delta_{i}]_{i=1,\dots,n}$ also belongs to
the kernel and is non-zero. Note that each $\delta_{i}$ can be written as

$\delta_{i}=P_{i}(x_{1},\dots,x_{n-1})$

where $P_{i}$ is an element of $T^{n-1}V$ depending only on $n$ and $i$. Now
if we consider

$P:=\sum_{i=1}^{n}P_{i}(x_{1},\dots,x_{n-1})e_{i}$

then $P$ is equal to $Q$ scaled by some factor. This shows that it satisfies
$P|_{F}\equiv 0$. Note that this equation stays true for any choice of
$x_{1},\dots,x_{n-1}$ for which the matrix $M$ is of rank $n-1$, and for such
a choice $P$ is a non-zero element of $V$. On the other hand, if
$x_{1},\dots,x_{n-1}$ are chosen in such a way that the rank of $M$ is smaller
than $n-1$ then $P\equiv 0$ so the equation $P|_{F}\equiv 0$ stays true. ∎

###  1.2 Proof of Theorem A

###### Proof of Theorem A.

Let $S:=X\times\mathbb{K}$ and let $V$ be the vector space of all polynomials
on $X\times\mathbb{K}$ which are of degree at most one with respect to the
second variable (i.e. of functions of the form $P_{1}(x)+P_{2}(x)t$, where
$P_{1},P_{2}\in\mathcal{P}(X)$). Fix some basis $(e_{i})_{i\in\mathbb{N}}$ of
$V$. For $Q\in V\backslash\\{0\\}$ define $c(Q)$ as in Lemma 1.1 and for a
subset $F\subset X\times K$ define $d(F)$ also as in Lemma 1.1.

Let $f:X\times Y\rightarrow\mathbb{K}$ be a mapping satisfying the assumptions
of Theorem A. By assumption $Y$ is algebraically of second category, so there
exists a natural number $n$ such that the set

$B:=\\{y\in Y:c(\Gamma_{f(\cdot,y)})=n\\}$

is Zariski dense in $Y$. Fix $y_{0}\in B$. Applying Lemma 1.1 with
$F=\Gamma_{f(\cdot,y_{0})}$, we can find polynomials $(P_{i})_{i=1,\dots,n}$
in $\mathcal{P}((X\times\mathbb{K})^{n-1})$ such that for all points
$x_{1},\dots,x_{n-1}\in X$ the polynomial

$P_{y_{0}}:=\sum_{i=1}^{n}P_{i}(x_{1},f(x_{1},y_{0}),\dots,x_{n-1},f(x_{n-1},y_{0}))e_{i}\in
V$

satisfies $P_{y_{0}}(x,f(x,y_{0}))=0$ for $x\in X$. Now the set of tuples
$(x_{1},\dots,x_{n-1})\in X^{n-1}$ for which $P_{y_{0}}$ is non-trivial is
Zariski open in $X^{n-1}$ and by assumption is non-empty, so we can find such
a tuple which belongs to $A^{n-1}$. Fix such a choice of
$(x_{1},\dots,x_{n-1})$.

The equation $P_{y}(x,f(x,y))=0$ by construction holds for every $(x,y)\in
X\times B$. As the coefficients of $P_{y}$ are regular functions in the
variable $y$, fixing $x\in A$ we conclude that $P_{y}(x,f(x,y))=0$ holds for
all $(x,y)\in A\times Y$. Now fixing $y$ we see that it holds for all
$(x,y)\in X\times Y$. This shows that

$Q(x,y,t):=P_{y}(x,t)$

is a non-trivial regular mapping of three variables $(x,y,t)\in X\times
Y\times\mathbb{K}$ which vanishes on the graph of $f$ and is of degree at most
one with respect to $t$. This gives the desired rational representation of
$f$. ∎

###  1.3 Proof of Theorem B

Throughout this section let $X,Y$ be two semialgebraic open subsets of
$R^{n_{X}}$ and $R^{n_{Y}}$ respectively for some $n_{X},n_{Y}\geq 0$, with
$X$ semialgebraically connected, as in Theorem B. First we need to repeat the
proof of Theorem A to arrive with a similar result in the Nash case.

######  Lemma 1.2.

Let $f:X\times Y\rightarrow R$ satisfy the assumptions of Theorem B. Then the
graph of $f$ is contained in a semialgebraic subset of $X\times Y\times R$ of
codimension one.

###### Proof.

By the semialgebraic cell decomposition ([1, Proposition 9.1.8]) $Y$ can be
written as a finite disjoint union of semialgebraic subsets, each Nash
diffeomorphic to $R^{n}$ for some $n\leq n_{Y}$. Hence without loss of
generality we might assume that $Y=R^{n_{Y}}$.

Let $S:=X\times R$ and let $V$ be the vector space of all mappings on $X\times
R$ which are restrictions of polynomials from $R^{n_{X}+1}$. Fix some basis
$(e_{i})_{i\in\mathbb{N}}$ of $V$. For $Q\in V\backslash\\{0\\}$ define $c(Q)$
as in Lemma 1.1 and for a subset $F\subset X\times K$ define $d(F)$ also as in
Lemma 1.1.

As for every $y\in Y$ the mapping $f(\cdot,y)$ is Nash we have that
$c(f(\cdot,y))<\infty$, so by Example 0.1 there exists a natural number $n$
such that the set

$B:=\\{y\in Y:c(\Gamma_{f(\cdot,y)})=n\\}$

is Zariski dense in $Y$. Fix $y_{0}\in B$. Applying Lemma 1.1 with
$F=\Gamma_{f(\cdot,y_{0})}$, we find polynomial functions
$(P_{i})_{i=1,\dots,n}\in\mathcal{P}(R^{(n_{X}+1)(n-1)})$ such that for all
points $x_{1},\dots,x_{n-1}\in X$ the mapping

$P_{y_{0}}:=\sum_{i=1}^{n}P_{i}(x_{1},f(x_{1},y_{0}),\dots,x_{n-1},f(x_{n-1},y_{0}))e_{i}\in
V$

satisfies $P_{y_{0}}(x,f(x,y_{0}))=0$ for $x\in X$. Now the set of tuples
$(x_{1},\dots,x_{n-1})\in X^{n-1}$ for which $P_{y_{0}}$ is the trivial
polynomial is the zero set of some Nash mapping on $X^{n-1}$ and by assumption
is non-empty. This means that we can find a tuple which belongs to $A^{n-1}$
for which $P_{y_{0}}$ is non-trivial. Fix such a choice of
$(x_{1},\dots,x_{n-1})$.

The equation $P_{y}(x,f(x,y))=0$ by construction holds for every $(x,y)\in
X\times B$. As the coefficients of $P_{y}$ are Nash functions in the variable
$y$, fixing $x\in A$ we conclude that $P_{y}(x,f(x,y))=0$ holds for all
$(x,y)\in A\times Y$. Now fixing $y$ we see that it holds for all $(x,y)\in
X\times Y$. This shows that

$Q(x,y,t):=P_{y}(x,t)$

is a non-trivial Nash mapping of three variables $(x,y,t)\in X\times Y\times
R$ which vanishes on the graph of $f$. As $X\times Y\times R$ is
semialgebraically connected, the zero set of $Q$ is the desired set of
codimension one containing $\Gamma_{f}$. ∎

Now we will introduce a few auxiliary results which will allow us to show that
Theorem B follows from Lemma 1.2.

######  Observation 1.3.

Let $U\subset X\times Y$ be a semialgebraic set such that
$U\cap(X\times\\{y\\})$ is dense in $X\times\\{y\\}$ for every $y\in Y$. Let
$g:U\rightarrow R$ be a semialgebraic mapping. For $y_{0}\in Y$ we define
$g(\cdot,y_{0})$ as the mapping $x\mapsto g(x,y_{0})$ defined on the set
$\\{x\in X:(x,y_{0})\in U\\}$. Then the set

$Y_{0}:=\\{y\in Y:g(\cdot,y)\textnormal{ can be extended to a continuous
function on }X\\}$

is semialgebraic and moreover the unique extension of $g|_{U\cap(X\times
Y_{0})}$ to $X\times Y_{0}$ which is continuous with respect to the first
variable is semialgebraic.

###### Proof.

Consider the embedding $R\subset\mathbb{P}^{1}(R)=R\cup\\{\infty\\}$ in the
projective space. As the operation of taking the closure of a set is
definable, the following set is semialgebraic:

$G:=\\{(x,y,t)\in X\times
Y\times\mathbb{P}^{1}(R):(x,t)\in\overline{\Gamma_{f(\cdot,y)}}\\}$

Now $Y_{0}$ can be written as

$Y_{0}=\\{y\in Y:\forall_{x\in X}(x,y,\infty)\not\in G\wedge\exists!_{t\in
R}\,(x,y,t)\in G\\}$

so it is semialgebraic. Finally the graph of the extension of
$g|_{U\cap(X\times Y_{0})}$ is precisely $G\cap(X\times Y_{0}\times R)$ so the
function is semialgebraic. ∎

######  Proposition 1.4.

Let $g:X\times Y\rightarrow R$ be a semialgebraic function, continuous with
respect to the variable $x\in X$. Then the set of points $y\in Y$ for which
$g(\cdot,y)$ is Nash is a semialgebraic subset of $Y$.

###### Proof.

According to [1, Theorem 8.10.5] for a fixed $g$ there exists a natural number
$r$ such that for $y\in Y$, $f(\cdot,y)$ is Nash if and only if it is of class
$\mathcal{C}^{r}$. This condition can easily be verified by a first order
formula as it depends only on finitely many partial derivatives of
$f(\cdot,y)$. ∎

###### Proof of Theorem B.

Let $f:X\times Y\rightarrow R$ satisfy the assumptions of Theorem B. We will
use induction on the dimension of $Y$, the case $n_{Y}=0$ being trivial. Let
$Z\subset Y$ be a semialgebraic closed subset of $Y$ of codimension at least
one. Thanks to the cell decomposition ([1, Proposition 9.1.8]), $Z$ can be
written as a disjoint union of its semialgebraic subsets, each of them Nash
diffeomorphic to $R^{n}$ for some $n<n_{Y}$. Now applying the induction
hypothesis to each of the cells separately we get that $f|_{X\times Z}$ is
semialgebraic. This means that every time in the proof we encounter a
semialgebraic subset of $Y$ of codimension at least one, relaying on the
induction hypothesis we can delete its closure $Z$ from $Y$ and consider
$Y^{\prime}=Y\backslash Z$ instead.

Let $F\subset X\times Y\times R$ be a semialgebraic set of codimension one
containing the graph of $f$, which exists by Lemma 1.2. By the cell
decomposition ([1, Proposition 9.1.12]), we can find a finite family
$(U_{i})_{i=1,\dots,n}$ of semialgebraic open disjoint sets covering a dense
subset of $X\times Y$ that for each $1\leq i\leq n$ there exists a finite
number of semialgebraic continuous mappings
$(\xi_{i,j})_{j=1,\dots,n_{i}}:U_{i}\rightarrow R$ such that for $(x,y)\in
U_{i}$ we have

$\displaystyle 1\leq j_{1}<j_{2}\leq
n_{i}\implies\xi_{i,j_{1}}(x)<\xi_{i,j_{2}}(x)$ $\displaystyle\text{for }t\in
R,(x,y,t)\in F\iff t=\xi_{i,j}(x,y)\text{ for some }j$

Subdividing the family further, by [1, Lemma 8.10.6] we can assume that for
each $y\in Y$ and $1\leq i\leq n$ we have that the set
$U_{i}\cap(X\times\\{y\\})$ is either empty or semialgebraically connected.
Define $U:=\bigcup U_{i}$ and $C:=X\times Y\backslash U$.

Consider

$\\{y\in Y:\dim(C\cap(X\times\\{y\\}))=n_{X}\\}$

It is a semialgebraic subset of $Y$ of codimension at least one. Relying on
the induction hypothesis, as stated at the beginning of the proof, we might
delete its closure from $Y$ to assume that

$\text{ for every }y\in Y,\;U\cap(X\times\\{y\\})\text{ is dense in
}X\times\\{y\\}$

Similarly now consider

$\\{x\in X:\dim(C\cap(\\{x\\}\times Y))=n_{Y}\\}$

it is a semialgebraic subset of $X$ of codimension at least one. This means
that we can find some $x_{0}\in A$ such that

$\dim(C\cap(\\{x_{0}\\}\times Y))<n_{Y}$

Again relying on the induction hypothesis, we can delete the set $\\{y\in
Y:(x_{0},y)\in C\\}$ from $Y$ to assume that

$C\cap(\\{x_{0}\\}\times Y)=\emptyset$ (1)

Now for every collection of natural numbers $K=(k_{i})_{i=1,\dots,n}$
satisfying $1\leq k_{i}\leq n_{i}$ for every $i$ define the semialgebraic
function $g_{K}:U\rightarrow R$ by

$g_{K}(x,y):=\begin{cases}\xi_{1,k_{1}}(x,y)&\text{ for }(x,y)\in U_{1}\\\
\dots\\\ \xi_{n,k_{n}}(x,y)&\text{ for }(x,y)\in U_{n}\end{cases}$

Consider

$B_{K}:=\\{y\in Y:g_{K}(\cdot,y)\text{ can be extended to a continuous
function on X}\\}$

By Observation 1.3 the set is semialgebraic. Let $\tilde{g}_{K}$ denote the
extension of $g_{K}$ to $X\times B_{k}$ which is continuous with respect to
the variable $x$. Again by Observation 1.3 the function is semialgebraic.
Finally consider:

$\widetilde{B}_{K}:=\\{y\in Y:\tilde{g}_{K}(\cdot,y)\text{ is Nash and
}\tilde{g}_{K}(x_{0},y)=f(x_{0},y)\\}$

By Proposition 1.4 the set is semialgebraic.

Fix now $y_{0}\in Y$. As the mapping $f(\cdot,y_{0})$ is continuous and the
sets $U_{i}\cap(X\times\\{y_{0}\\})$ are semialgebraically connected, for
every $i$ we must have that there exists $1\leq j\leq n_{i}$ such that
$f(x,y_{0})=\xi_{i,j}(x,y_{0})$ holds for $x$ with $(x,y_{0})\in U_{i}$. This
implies that there exists a multiindex $K$ such that
$y_{0}\in\widetilde{B}_{k}$ and $\tilde{g}_{K}(x,y_{0})=f(x,y_{0})$ holds for
$x\in X$.

This in particular shows that the sets $\widetilde{B}_{K}$ cover the entire
$Y$. Note that if $y\in\widetilde{B}_{K}\cap\widetilde{B}_{K^{\prime}}$ for
some two multiindices $K$ and $K^{\prime}$ then

$\tilde{g}_{K}(x,y)=\tilde{g}_{K^{\prime}}(x,y)\text{ for }x\in X$

as both of the mappings are Nash with respect to $x$ and agree on a
neighbourhood of $x_{0}$ (thanks to (1)). This shows that $f$ can be written
as

$f(x,y)=\begin{cases}\tilde{g}_{K_{1}}(x,y)&\text{ for
}y\in\tilde{B}_{K_{1}}\\\ \dots\\\ \tilde{g}_{K_{N}}(x,y)&\text{ for
}y\in\tilde{B}_{K_{N}}\end{cases}$

where $(K_{i})_{i=1,\dots,N}$ ranges through all the possible multiindices.
This proves that $f$ is semialgebraic. ∎

## References

  * [1] Jacek Bochnak, Michel Coste, and Marie-Françoise Roy.  Real Algebraic Geometry.  Springer Berlin Heidelberg, Berlin, Heidelberg, 1998.
  * [2] Beata Gryszka and Janusz Gwoździewicz.  On some regularity condition, 2021.
  * [3] Wojciech Kucharz, Krzysztof Kurdyka, and Ali El-Siblani.  Separately nash and arc-nash functions over real closed fields.  Bulletin of the London Mathematical Society, 53(2):426–441, 2021.
  * [4] Richard S. Palais.  Some analogues of Hartogs’ Theorem in an algebraic setting.  American Journal of Mathematics, 100(2):387–405, 1978.

Juliusz Banecki, Faculty of Mathematics and Computer Science, Jagiellonian
University, ul. Lojasiewicza 6, 30-348 Krakow, Poland
E-mail address<EMAIL_ADDRESS>