(1) The coldest objects are rare: Our result that the mass function is best
fit with a power law of $\alpha=0.6$ and that the cutoff mass is likely at or
below 5$M_{Jup}$ would imply a distribution of objects in the 150-300K bin
like that shown in the green curve in the lower right panel of Figure 29b.
This implies a space density of at least $2{\times}10^{-3}pc^{-3}$, which
makes objects in this bin as common as T6 or T7 dwarfs. It is thus hard to
reconcile these results with the hypothesis that such cold objects are
extremely rare. Furthermore, it would be an unbelievable stroke of
luck222222It is already an oddity that our G star has, as its four closest
neighbors, systems that harbor 1 G dwarf, 1 K dwarf, 2 M dwarfs, 1 L dwarf, 1
T dwarf, and 1 Y dwarf, since a random draw of the overall mass function would
be heavily weighted toward M dwarfs plus a random K or T dwarf but weighted
against rarer G or L dwarfs. See Kirkpatrick et al. 2012 for the full-sky 8-pc
sample. that our Sun falls a mere 2.3 pc from such an extremely rare, cold
object, as it does with WISE 0855$-$0714\. So we reject rarity as a possible
cause.

(2) WISE is too shallow: History has shown us that all-sky surveys can lead to
curious results when researchers push those surveys near their limits. The
bottom of the main sequence in the 1980s appeared to fall at late-M (Probst &
Liebert 1983; Reid 1987) based on the dominant discovery engine of its time,
the Palomar Observatory Sky Survey (Minkowski & Abell 1963; Reid et al. 1991).
We now know, of course, that the reason for this is the low space density of
early-L dwarfs (see Figure 27) and the fact that the POSS-I and POSS-II $B$
and $R$ plates failed to survey enough volume to detect all but the nearest L
dwarf examples. The L/T pair WISE 1049$-$5319 is present on the southern UK
Schmidt photographic plates but was not selected as a motion source (Luhman
2013); we find that Willem Luyten, despite having cataloged over 58,000 proper
motion stars using photographic data (Luyten 1979), failed to catalog any of
the 20-pc L dwarfs in Table 11. In the case of WISE, Wright et al. (2014) have
used the relatively bright W2 magnitude of WISE 0855$-$0714 (W2 = 13.82 mag),
its distance (2.3 pc), and the fact that it lies $\sim$2 magnitudes above the
limit of the AllWISE Catalog to argue that there should be another 4 to 35
similar objects already detected in AllWISE itself. The CatWISE and
CatWISE2020 Catalogs (see below) have increased the sensitivity to lower
motions at fainter magnitudes, thus making the identification of these
detected objects even easier. Hence, it is unlikely that the survey that found
WISE 0855$-$0714 is too shallow to find other analogs.

(3) High motions confound deeper searches: The data sets using the longest
time baseline of WISE data are CatWISE Preliminary (Eisenhardt et al. 2020)
and CatWISE2020 (Marocco et al. 2020b). Most points on the celestial sphere
are visited by WISE during a several-day window every six months. Both the
CatWISE Preliminary and CatWISE2020 processing leveraged these repeats to
measure proper motions of all sources. Full-depth coadditions, which took all
of the available data to create a single, deep image, were used for source
detection. Those source detections were then characterized through the stack
of epochal coadds (from each six-month window) to measure photometry and
astrometry for each source. Sources with significant proper motions could then
be selected from the resulting source tables. Sources that fail to move a
significant portion of a full-depth coadd’s W2 FWHM (${\sim}6\arcsec$; Meisner
et al. 2019) benefit from the coaddition, as their S/N increases by roughly
the square root of the number of epochs. However, sources with higher motions
do not see this benefit; a very high motion source will appear as a tracklet
of separate sources in the full-depth coadd, and each separate apparition
contains the background noise component from all epochs but the source signal
from only one. Therefore, faint, high-motion sources can be lost in this
process. If many of the coldest brown dwarfs are older kinematically, as
Figure 29a and b suggest, their concomitant high proper motions may quash
their identification by the CatWISE pipeline.

(4) Cold objects have unexpected colors or magnitudes: The analysis from
Wright et al. (2014) inherently assumed that WISE 0855$-$0714 is a
representative member of the Y dwarfs populating the 150-300K bin. What if
WISE 0855$-$0714 is atypical? It has $v_{tan}=88.0$ km s-1, which, although in
the highest 4% of all $v_{tan}$ values in Figure 31, is not exceptional. If
the majority of objects in the 150-300K bin are much older and have higher
kinematics, then their high motions may suggest that point (3) above is a
contributing cause. In addition, however, their older ages would also suggest
a somewhat lower metallicity in general. If we look at the 20-pc T subdwarfs
(section 7.2) that have metallicity measurements, we find that values as low
as $[M/H]=-0.3$ dex produce noticeable changes in the spectra of mid- to
late-T dwarfs. Values of $[M/H]=-0.6$ dex begin to move objects into
unfamiliar loci on color-magnitude diagrams. Inasmuch as molecular absorption
strengths dictate the overall spectral energy distribution of Y dwarfs (Figure
15 of Doré et al. 2016), slight changes in metallicity could affect the
relative importance of these bands and dramatically alter Y dwarf spectra and
colors. Recent discoveries at early-T from Schneider et al. (2020) and Meisner
et al. (2021) underscore the point that warmer brown dwarfs with presumably
lower metallicity ([Fe/H] $\leq-1$ dex) exist; their spectra are vastly
different, at least in the near-infrared, from those of solar-metallicity T
dwarfs. These may be harbingers of the photometric and spectroscopic
bizarreness we can expect from the majority of later Y dwarfs, even if these Y
dwarfs in general have less extreme metallicities.

In summary, other nearby objects with temperatures comparable to WISE
0855$-$0714 must exist, based on evidence from the mass function shape and
knowledge of its low-mass cutoff. However, the expected higher motions and
lower metallicities of objects in this 150-300K bin, may make them a challenge
to identify, especially when coupled with their intrinsic faintness.

Figure 31: Tangential velocities plotted against effective temperature for L,
T and Y dwarfs in the 20-pc census. Only those objects having parallax
measurements with uncertainties $<$12.5% are shown. Individual objects are
shown as black squares and the median $v_{tan}$ values in each 150K bin are
shown as white triangles.

##  10 Conclusions

Our results, which use the final trigonometric parallaxes we have measured
using Spitzer, confirm the result of Kirkpatrick et al. (2019) that the 20-pc
brown dwarf portion of the mass function, which is based here on 525 L through
Y dwarfs, can be best described as a power law with an exponent of
$\alpha=0.6{\pm}0.1$. We have not yet, however, extended this analysis to
higher masses to investigate how the mass function behaves over the entire
mass range within 20 pc. Earlier analyses have indicated that the higher mass
portion can be described as a two-part power law (Kroupa et al. 2013) or log-
normal form (Chabrier 2003b). New data, particularly data from Gaia DR2 and
subsequent releases can be used to refine our knowledge of the A through M
dwarfs (and white dwarfs) with the 20-pc census as well as providing important
astrometric information to help identify companions to those stars. Developing
a database containing all knowledge of our stellar and substellar neighbors
within this volume will enable us to explore the individual-object mass
function with unprecedented detail.

Our results have also shown that the cutoff mass for star formation, is
constrained to be lower than ${\sim}10M_{Jup}$ and that analysis of young
moving group members over a wider sample likely constrains this value to
${\sim}5M_{Jup}$. Obtaining a more solid value for the cutoff mass requires
volume-complete subsets of a substantial number of Y dwarfs colder than 450K,
and particularly below $\sim$350K, a regime in which we have only one
confirmed Y dwarf. Although WISE has provided a trove of Y dwarf discoveries,
probing a substantial volume colder than $\sim$350K may require other
resources. One such resource currently being planned is the Near Earth Object
Surveyor (formerly called NEOCam) that is due to launch in 2025. As discussed
in Kirkpatrick et al. (2019b), NEO Surveyor will cover 64% of the celestial
sphere in two bands, NC1 and NC2, that cover wavelengths of 4.0-5.2 $\mu$m and
6.0-10.0 $\mu$m. Portions of the sky will be repeatedly scanned during their
75-day visibility windows then scanned again roughly 215 days later when the
next visibility window opens. The mission, although planned for five years,
has a design lifetime of twelve years.

The absolute NC1 fluxes of 350K Y dwarf and a 250K Y dwarf are 103 $\mu$Jy and
26 $\mu$Jy, respectively. The use of image differencing for high-motion
objects in NEO Surveyor data will theoretically allow us to achieve single-
epoch S/N=5 sensitivities of $\sim$4 $\mu$Jy at NC1, thereby greatly
increasing the distances to which we can detect these coldest brown dwarfs.
However, NEO Surveyor is run through NASA’s Planetary Defense Coordination
Office, so no funding is being provided for the additional processing needed
for astrophysical studies. For a relatively small investment, NASA
Astrophysics could realize the full potential of NEO Surveyor data for stellar
astrophysical research, of which cold brown dwarf discovery would be a major
beneficiary.

This publication makes use of data products from WISE, which is a joint
project of the University of California, Los Angeles, and the Jet Propulsion
Laboratory (JPL), California Institute of Technology (Caltech), funded by the
National Aeronautics and Space Administration (NASA). Work in this paper is
based on observations made with the Spitzer Space Telescope, which is operated
by JPL/Caltech, under a contract with NASA. Support for this work was provided
to Davy Kirkpatrick by NASA through a Cycle 14 award issued by JPL/Caltech.
Some of the data presented here were obtained at the W. M. Keck observatory,
which is operated as a scientific partnership among Caltech, the University of
California, and NASA. The Observatory was made possible by the generous
financial support of the W. M. Keck Foundation. The authors wish to recognize
and acknowledge the very significant cultural role and reverence that the
summit of Maunakea has always had within the indigenous Hawaiian community. We
are most fortunate to have the opportunity to conduct observations from this
mountain. Results here are partly based on observations obtained at the Hale
Telescope, Palomar Observatory, as part of a continuing collaboration between
Caltech, NASA/JPL, Yale University, and the National Astronomical
Observatories of China. We would like to thank SURF students Tea Freedman-
Susskind, Emily Zhang, Yerong Xu, and Feiyang Liu for help with the
spectroscopic observation of WISE 2126+2530 from Palomar. This work has made
use of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This research has made use
of IRSA, which is operated by JPL/Caltech, under contract with NASA. This
research has also made use of the SIMBAD database, operated at CDS,
Strasbourg, France. Federico Marocco acknowledges support from grant
#80NSSC20K0452 under the NASA Astrophysics Data Analysis Program. Alfred
Cayago gratefully acknowledges financial support through the Fellowships and
Internships in Extremely Large Data Sets (FIELDS) Program, a National
Aeronautics and Space Administration (NASA)
science/technology/engineering/math (STEM) grant administered by the
University of California, Riverside. Emily Martin is supported by an NSF
Astronomy and Astrophysics Postdoctoral Fellowship under award AST-1801978.
Eileen Gonzales acknowledges support from an LSSTC Data Science Fellowship.
Christopher Theissen acknowledges support for this work through NASA Hubble
Fellowship grant HST-HF2-51447.001-A awarded by the Space Telescope Science
Institute, which is operated by the Association of Universities for Research
in Astronomy, Inc., for NASA, under contract NAS5-26555. The Backyard Worlds:
Planet 9 team thanks Zooniverse volunteers who have participated in the
project. Backyard Worlds research was supported by NASA grant 2017-ADAP17-0067
and by the NSF under grants AST-2007068, AST-2009177, and AST-2009136. CatWISE
is led by JPL/Caltech, with funding from NASA’s Astrophysics Data Analysis
Program. This research was partly carried out at JPL/Caltech, under contract
with NASA. We thank the referee for a quick report despite difficulties
imposed by the current pandemic.

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##  Appendix A Spectral Types, Astrometry, and Photometry for Systems

For systems in Tables 5, 6, 7, 8, 9, 10, and 11, we have collected
spectroscopic, astrometric, and photometric data from both this paper and the
literature. These data are listed in Table 17. The various sections of the
table are described in detail below. Close binaries are generally entered as a
single entry with joint photometry unless there are components of the multiple
with spectral types earlier than L0. For a full accounting of individual L, T,
and Y components within the 20-pc census, refer to Table 11.

###  A.1 Origin and Name

Column $T$ indicates the table(s) from which the source originates. Objects in
the 20-pc census (Table 11) are indicated by "T". Users are encouraged to use
this column rather than the parallax column if they wish to select the same
set of objects that we included in our 20-pc census. Objects that are not
listed in our 20-pc census (Table 11) but were nonetheless part of our Spitzer
parallax program (Tables 5-7) are indicated by "P". Objects that are not from
any of these tables but were part of our photometric or spectroscopic follow-
up campaigns (Tables 8 and 9) are indicated by "F". Objects considered for the
20-pc census but ultimately not included (Table 10) are indicated by "C".

Column $ShortName$ gives the abbreviated prefix and suffix of the full source
name. This prefix is generally the survey of origin, and the abbreviated
suffix is the sexagesimal RA and Dec of the source in the form
$hhmm{\pm}ddmm$. As examples, CWISEP J193518.59$-$154620.3 is denoted as CWISE
1935$-$1546, and PSO J149.0341$-$14.7857 is denoted as PSO 0956$-$1447\.
Exceptions are made for objects with common names like Gl 570D and LHS 2397aB,
whose full names are used instead.

###  A.2 Spectral Types

Columns $SpO$ and $SpIR$ list the optical and near-infrared spectral types,
respectively, if known. These are converted to a decimal scale, and any
qualifying criteria such as "pec", "$\beta$", and "sd" are dropped. The
convention for the decimal scale is L0 = 0.0, T0 = 10.0, and Y0 = 20.0. As
examples, an object with a spectral type of sdT8 is given as 18.0, and one
with a type of L7: VL-G is given as 7.0. The two objects listed in Table 11
with types of "extremely red" in Mace et al. (2013) are given in this table as
9.5. Column $SpAd$ is the adopted spectral type, which is the same as $SpIR$
if that value is not null; otherwise, it is the same as $SpO$. If both of
those quantities are null, a spectral type estimate is given. A few objects,
however, have null values for $SpAd$, and these are objects believed to be
background interlopers and not brown dwarfs.

The source of the spectral type is given in column $OI$. An explanation of the
double-letter code for this column can be found in the table comments.

###  A.3 Astrometric Data

Columns $\varpi_{abs}$, $\mu_{\alpha}$, and $\mu_{\delta}$ list the best
measured trigonometric parallax and proper motion values in RA and Dec. The
"best" astrometry is simply that data set with the smallest quoted uncertainty
in the parallax or, for objects lacking a parallax measurement, the data set
with the smallest quoted uncertainty in the total proper motion. All
parallaxes are given on the absolute reference grid; data from Tinney et al.
(2003) and Tinney et al. (2014), along with USNO data from Kirkpatrick et al.
(2019), were converted from relative to absolute as described in section 8 of
Kirkpatrick et al. (2019). The values listed for proper motion are a mixture
of relative and absolute measurements. Readers are encouraged to cite the
source of those values if this distinction is important for their research.

The source of the astrometry is given in column $AS$. An explanation of the
single-letter code for this column can be found in the table comments.

###  A.4 $JHK$ Photometry

Column $J_{MKO}$ lists $J$-band photometry on the MKO system, $J_{2MASS}$
lists $J$-band photometry one the 2MASS system, $H$ lists $H$-band photometry
on either the MKO or 2MASS system, $K_{MKO}$ lists $K$-band photometry on the
MKO system, and $K_{S(2MASS)}$ lists $K_{S}$-band photometry on the 2MASS
system. See section 5.1.1 for details. Photometric values listed without
corresponding errors are magnitude limits.

The source of the photometry is given in column $PhotS$. An explanation of the
five-letter code for this column can be found in the table comments.

###  A.5 CatWISE2020 Data

Columns $RA\\_C2$, $Dec\\_C2$, $pmra\\_C2$, $pmdec\\_C2$, $W1mag\\_C2$,
$W2mag\\_C2$, and $par\\_C2$ contain astrometric information from the
CatWISE2020 Catalog and Reject Table (Marocco et al. 2020b). The first two
columns are the J2000 equinox RA and Dec positions from the moving-object
solution at epoch MJD 57170.0, the next two columns are the measured proper
motion and their uncertainties in RA and Dec, the next two columns are the
moving-object PSF-fit photometry in WISE bands W1 and W2, and the final column
is a crude measurement of the object’s parallax (called par_pm in the
documentation).

The source of the CatWISE2020 data is given in column $C2S$. Upper-case "C2"
refers to the Catalog and lower-case "c2" refers to the Reject Table.

###  A.6 AllWISE Data

Columns $W1mag$, $W2mag$, and $W3mag$ provide stationary-object PSF-fit
measurements (primarily from AllWISE) in WISE bands W1, W2, and W3. These are
provided for two reasons. First, CatWISE2020 does not provide any W3 data,
since this band was not available for the post-cryogenic phases of the WISE
and NEOWISE missions. Second, the short, six-month time baseline of AllWISE
means that this stationary-object photometry should be robust for all sources
except those of exceptionally large motion, and thus the W1 and W2 photometry
can be compared to the moving-object photometry from CatWISE2020 to provide
another photometric check.

The source of the stationary-object photometry is given in column $WS$. In
most cases, this is the AllWISE Source Catalog or Reject Table. Some sources,
however, were not detected until crowdsource (Schlafly et al. 2018) was used
on the unWISE images underlying the CatWISE2020 processing. In this case, the
stationary-object W1 and W2 photometry from CatWISE2020 is listed instead.

###  A.7 Spitzer Data

Columns $ch1mag$ and $ch2mag$ provide the Spitzer channel 1 (3.6 $\mu$m) and
channel 2 (4.5$\mu$m) photometry. The source of this photometry is given in
column $SS$, the single-character code for which is described in the table
comments.

###  A.8 Note and Full Designation

Column $Note$ lists a one-letter code indicating whether the object is an
unresolved multiple (M); a young, low-gravity object (Y), or an old, subdwarf
(S). Column $FullName$ gives the full discovery designation of the system.

=65mm

Table 17: Amassed Spectroscopic, Astrometric, and Photometric Data for Objects
Listed in Tables 5, 6, 7, 8, 9, 10, and 11

T | Name | SpO | SpIR | SpAd | OI | $\varpi_{abs}$ | $\mu_{\alpha}$ | $\mu_{\delta}$ | AS | $J_{MKO}$ | $J_{2MASS}$ | $H$ | $K_{MKO}$ | $K_{S(2MASS)}$ | PhotS | RA_C2 | Dec_C2 | …
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  | (mas) | (mas/yr) | (mas/yr) |  | (mag) | (mag) | (mag) | (mag) | (mag) |  | (deg) | (deg) |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13) | (14) | (15) | (16) | (17) | (18) | (19)
T | SDSS 0000+2554 | 15.0 | 14.5 | 14.5 | TT | 70.8 $\pm$ 1.9 |  $-$19.1 $\pm$ 1.5 | 126.7 $\pm$ 1.3 | D | 14.85$\pm$0.01 | 15.06$\pm$0.04 | 14.73$\pm$0.07 | 14.82$\pm$0.03 | 14.84$\pm$0.12 | U22D2 | 0.0563043 | 25.9054854 |
T | GJ 1001BC | 5.0 | 5.0 | 5.0 | kT | 82.0946$\pm$ 0.3768 | 671.09$\pm$ 0.35 |  $-$1498.16$\pm$ 0.51 | G | 12.98$\pm$0.01 | 13.11$\pm$0.02 | 12.06$\pm$0.03 |  $\cdots$ | 11.39$\pm$0.01 | V22-V | 1.1491771 |  $-$40.7415963 |
T | WISE 0005+3737 |  $\cdots$ | 19.0 | 19.0 | -T | 126.9 $\pm$ 2.1 | 997.3 $\pm$ 1.0 |  $-$271.6 $\pm$ 1.0 | T | 17.58$\pm$0.04 |  $\cdots$ | 17.98$\pm$0.02 |  $\cdots$ | 16.28$\pm$0.31 | U-k-2 | 1.3250452 | 37.6219054 |
T | 2MASS 0014$-$4844 | 2.5 | 2.5 | 2.5 | TT | 50.1064$\pm$ 0.3898 | 870.72$\pm$ 0.27 | 281.46$\pm$ 0.43 | G | 13.91$\pm$0.01 | 14.05$\pm$0.04 | 13.26$\pm$0.01 |  $\cdots$ | 12.78$\pm$0.01 | V2V-V | 3.7386552 |  $-$48.7367024 |
T | WISE 0015$-$4615 |  $\cdots$ | 18.0 | 18.0 | -T | 75.2 $\pm$ 2.4 | 413.4 $\pm$ 1.1 |  $-$687.8 $\pm$ 1.0 | T | 17.67$\pm$0.02 |  $\cdots$ | 17.91$\pm$0.07 |  $\cdots$ |  $\cdots$ | V-V– | 3.7755685 |  $-$46.2558784 |
T | 2MASS 0015+3516 | 2.0 | 1.0 | 1.0 | TT | 58.6085$\pm$ 0.3664 | 55.17$\pm$ 0.45 |  $-$257.09$\pm$ 0.28 | G | 13.71$\pm$0.01 | 13.88$\pm$0.03 | 12.89$\pm$0.04 |  $\cdots$ | 12.26$\pm$0.02 | UKK-2 | 3.9368016 | 35.2663391 |

Note. — Only a portion of the table’s rows and columns is shown here. The full
table is available from the journal in machine-readable format.

Note. — References for OI, where the reference for the optical (O) spectral
type is given as the first character and that for the near-infrared (I)
spectral type is given as the second character: (a) Albert et al. 2011, (A)
Thompson et al. 2013, (b) Burningham et al. 2010, (B) Burgasser, et al. 2010b,
(c) Cushing et al. 2011, (C) Cushing et al. 2018, (d) Kirkpatrick et al. 2012,
(D) Kirkpatrick et al. 2000, (e) Martin et al. 2018, (E) Reid et al. 2001b,
(f) Kirkpatrick et al. 2010, (F) Faherty et al. 2014, (g) Burgasser et al.
2006, (G) Bardalez Gagliuffi, et al. 2014, (h) Hawley, et al. 2002, (H) Dhital
et al. 2011, (i) Chiu et al. 2006, (I) Koen, et al. 2017, (J) Kirkpatrick et
al. 1999, (j) Kirkpatrick et al. 2008, (k) Kirkpatrick et al. 2001, (K)
Kirkpatrick et al. 2011, (l) Kendall et al. 2007, (L) Kendall et al. 2003, (m)
Artigau et al. 2011, (M) Mace et al. 2013, (n) Scholz et al. 2003, (N) King et
al. 2010, (p) Potter, et al. 2002, (P) Pineda et al. 2016, (q) Gizis 2002, (Q)
Cruz, et al. 2007, (r) Reid et al. 2008, (R) Reid, et al. 2006, (s) Schneider,
et al. 2014, (S) Schneider et al. 2015, (t) Tinney et al. 2018, (T) See Tables
9-12 in this paper for references, (u) Burningham et al. 2013, (U) Burgasser
2007, (v) Schneider et al. 2017, (V) Kirkpatrick et al. 2016, (w) Kendall, et
al. 2004, (W) Best et al. 2013, (X) Burgasser et al. 2003d, (x) Thorstensen &
Kirkpatrick 2003, (y) Deacon et al. 2014, (Y) Reylé et al. 2014, (z) Burgasser
et al. 2010, (Z) Fan, et al. 2000.

Note. — References for AS, the source of the astrometric data: (A) Dahn et al.
2002, (b) Burgasser et al. 2008a, (B) Bartlett et al. 2017, (c) CatWISE2020
Catalog, (C) Tinney et al. 2014, (d) Dahn et al. 2017, (D) Dupuy & Liu 2012,
(E) Dupuy et al. 2019, (F) Faherty et al. 2012, (G) Gaia DR2 - quoted
astrometry is for the actual source listed, (g) Gaia DR2 - quoted astrometry
is that of the brighter primary in the system, (J) Kirkpatrick et al. 2011,
(H) Hipparcos - van Leeuwen 2007, (K) Kirkpatrick et al. 2019 for NTT and
UKIRT parallaxes, (k) Kirkpatrick et al. 2019 for USNO parallaxes, (l) Leggett
et al. 2012, (L) Liu et al. 2016, (m) Manjavacas et al. 2013, (M) Marocco et
al. 2010, (r) Smart, priv. comm., (R) Smart et al. 2018, (S) Casewell et al.
2008, (s) Smart et al. 2013, (t) Tinney et al. 2003, (T) This paper, (V) Vrba
et al. 2004, (W) Best et al. 2020, (z) Dupuy et al. 2020, (Z) Lazorenko, &
Sahlmann 2018.

Note. — References for PhotS, the source of the $J,H,K$ photometry: (2) 2MASS
Skrutskie et al. 2006, (a) Meisner et al. 2020a, (A) Meisner et al. 2020b, (b)
Bardalez Gagliuffi et al. 2020: Note that the HST F125W magnitude limit for
WISE 0830+2837 is used as its value for $J_{MKO}$, (B) Bigelow/2MASS, (c)
Boccaletti et al. 2003, (C) CTIO-4m/NEWFIRM, (d) Kirkpatrick et al. 2012, (D)
Database of Ultracool Parallaxes as of 2020 April: Dupuy & Liu 2012, Dupuy &
Kraus 2013, and Liu et al. 2016, (e) Martin et al. 2018, (E) McElwain &
Burgasser 2006, (f) Faherty et al. 2012, (F) Freed et al. 2003, (g) Mamajek et
al. 2018, (G) Gemini-South/FLAMINGOS2, (h) Pinfield et al. 2014b, (H) Pinfield
et al. 2014a, (i) Ireland et al. 2008, (I) Dupuy et al. 2019, (j) Janson et
al. 2011, (J) Faherty et al. 2014b, (k) Kirkpatrick et al. 2019, (K)
Kirkpatrick et al. 2011, (m) Mace et al. 2013, (M) Magellan/PANIC, (p)
PAIRITEL, (P) Palomar/WIRC, (q) Dhital et al. 2011, (Q) Deacon et al. 2017a,
(r) Deacon et al. 2012b, (s) Schneider et al. 2015, (S) SOAR/OSIRIS, (t)
Tinney et al. 2014, (T) Thompson et al. 2013, (u) ULAS, UGPS, or UGCS, (U)
UHS, (v) VVV, (V) VHS, (w) Wright et al. 2013, (W) Best et al. 2020.

Note. — References for C2S, the source of the CatWISE2020 data: (C2)
CatWISE2020 Catalog, (c2) CatWISE2020 Reject Table.

Note. — References for WS, the source of the WISE photometry: (AW) AllWISE
Source Catalog, (aw) AllWISE Reject Table, (C2) CatWISE2020 Catalog, (c2)
CatWISE2020 Reject Table.

Note. — References for SS, the source of the Spitzer photometry: (0) This
paper, (f) Marocco et al. 2020, (F) Filippazzo, et al. 2015, (K) Kirkpatrick
et al. 2019, (L) Leggett et al. 2007, (M) Meisner et al. 2020a, (m) Meisner et
al. 2020b, (P) Patten et al. 2006, (S) Metchev et al. 2015.


#

Santiago Laplagne  Instituto de Cálculo, FCEN, Universidad de Buenos Aires -
Ciudad Universitaria, Pabellón I - (C1428EGA) - Buenos Aires, Argentina
<EMAIL_ADDRESS>
# On the Pythagoras number for polynomials of degree 4 in 5 variables

Santiago Laplagne  Instituto de Cálculo, FCEN, Universidad de Buenos Aires -
Ciudad Universitaria, Pabellón I - (C1428EGA) - Buenos Aires, Argentina
<EMAIL_ADDRESS>
###### Abstract.

We give an example of a polynomial of degree 4 in 5 variables that is the sum
of squares of 8 polynomials and cannot be decomposed as the sum of 7 squares.
This improves the current existing lower bound of 7 polynomials for the
Pythagoras number $p(5,4)$.

##  1\. Introduction

The decomposition of a real multivariate polynomial as a sum of squares of
real polynomials is a central problem in real algebraic geometry, with many
theoretical and practical applications. An interesting and difficult problem
is to determine the minimum number $p(n,2d)$ such that any polynomial of
degree $2d$ in $n$ variables that is a sum of squares can be decomposed as a
sum of $p(n,2d)$ squares. This number is called the Pythagoras number of
$n$-ary forms of degree $2d$.

In [8], C. Scheiderer obtained lower bounds for $p(n,2d)$ for all $n\geq 3$
and $d\geq 2$, assuming a conjecture by Iarrovino and Kanev that can be
verified computationally for small values of $n$ and $d$. The lower bounds
obtained in that paper are close to the known upper bounds for the Pythagoras
number. A natural question is whether these lower bounds are sharp. For the
case of polynomials of degree 4 in 5 variables, the lower bound given in [8]
is 7. That is, there exists a polynomial that is the sum of 7 squares and
cannot be decomposed as the sum of 6 squares. In this paper, we give an
explicit example that improves this bound. Our polynomial is a sum of 8
squares and cannot be decomposed as the sum of 7 squares. As far as we know,
this is the first example that improves the lower bounds given in [8] for any
$n$ and $d$, showing that the examples constructed there are not always
optimal.

###  1.1. Preliminaries

We set some notations and recall basic results that will be used in our
construction. We refer the readers to [2, Chapter 3 and 4] and [4] for details
and proofs.

####  1.1.1. Sums of squares

Let $H_{n,d}$ be the space of homogeneous polynomials of degree $d$ in $n$
variables and let $\Sigma_{n,2d}\subset H_{n,2d}$ be the set of polynomials
that can be decomposed as a sum of squares of polynomials, which we call
shortly SOS polynomials. The set $\Sigma_{n,2d}$ is a full dimensional cone in
$H_{n,2d}$, included in $P_{n,2d}$, the cone of non-negative polynomials in
$H_{n,2d}$. If $f\in\Sigma_{n,2d}$, $f=p_{1}^{2}+\dots+p_{s}^{2}$, then
$p_{i}\in H_{n,d}$ for all $1\leq i\leq s$ (that is, all the polynomials in
the SOS decomposition of a homogeneous polynomial of degree $2d$ are
homogeneous polynomials of degree $d$).

####  1.1.2. The Gram Spectrahedron

Let $m$ be the vector of monomials of degree $d$ in $n$ variables under some
monomial ordering and let $N$ be the length of this vector ($N=\dim H_{n,d}$).
A homogeneous polynomial $f\in\mathbb{R}[x_{1},\dots,x_{n}]$ of degree $2d$ is
a sum of squares if and only if there exists a positive semidefinite matrix
$A\in\mathbb{R}^{N\times N}$ such that

$f=m^{T}Am.$

The space of all such matrices is called the Gram spectrahedron of $f$. It is
a compact convex set in the space of matrices.

####  1.1.3. The dual cone and bilinear forms

The dual cone $K^{*}$ of a convex cone $K$ in a real vector space $V$ is the
set of all linear functionals in the dual space $V^{*}$ that are nonnegative
on $K$: $K^{*}=\\{\ell\in V^{*}\,:\,\ell(x)\geq 0,\,\forall x\in K\\}$. Given
a form $\ell\in H_{n,2d}^{*}$, we can define a bilinear form
$Q_{\ell}:H_{n,d}\times H_{n,d}\rightarrow\mathbb{R}$,
$Q_{\ell}(p,q)=\ell(pq)$. For computations, it is convenient to fix a monomial
base $\mathcal{M}$ of $H_{n,d}$ and represent $Q_{\ell}$ in the coordinates of
$\mathcal{M}$ as a matrix $\mathsfit{Q}\in\mathbb{R}^{N\times N}$, where
$N=\dim H_{n,d}$. If $\ell\in\Sigma_{n,2d}^{*}$, then $Q_{\ell}(p,p)\geq 0$
for all $p\in H_{n,d}$. That is, the form $Q_{\ell}$ is positive semi-definite
and so is the matrix $\mathsfit{Q}$. The converse is also true: if $\ell\in
H_{n,2d}^{*}$ and $Q_{\ell}$ is positive semidefinite, then
$\ell\in\Sigma_{n,2d}^{*}$.

If $f$ is in the boundary of $\Sigma_{n,2d}$, there exists
$\ell\in\Sigma_{n,2d}^{*}$ such that $\ell(f)=0$. If
$f=p_{1}^{2}+\dots+p_{s}^{2}$ then $\ell(p_{i}^{2})=0$ for all $1\leq i\leq
s$. Therefore, $Q_{\ell}(p_{i},p_{i})=0$ and $(p_{i})_{\mathcal{M}}$ is in the
kernel of the matrix $\mathsfit{Q}$ for all $1\leq i\leq s$ (since
$\mathsfit{Q}$ is positive semidefinite).

##  2\. Sum of 8 polynomials

We give an example of a polynomial of degree 4 in 5 variables that is the sum
of 8 squares and cannot be decomposed as the sum of 7 squares.

###  2.1. Construction

Our starting point is a strictly positive polynomial of degree 4 in 4
variables in the boundary of $\Sigma_{4,4}$. In [1], G. Blekherman provides
formulas for constructing such polynomials. We use the example given in [6,
Example 4.3] following those formulas.

######  Example 2.1.

Let $p_{1},p_{2},p_{3},p_{4}\in\mathbb{R}[x_{1},x_{2},x_{3},x_{4}]$,

$\displaystyle p_{1}$ $\displaystyle=x_{1}^{2}-x_{4}^{2},$ $\displaystyle
p_{2}$ $\displaystyle=x_{2}^{2}-x_{4}^{2},$ $\displaystyle p_{3}$
$\displaystyle=x_{3}^{2}-x_{4}^{2},$ $\displaystyle p_{4}$
$\displaystyle=-x_{1}^{2}-x_{1}x_{2}-x_{1}x_{3}+x_{1}x_{4}-x_{2}x_{3}+x_{2}x_{4}+x_{3}x_{4},$

and set $g=p_{1}^{2}+p_{2}^{2}+p_{3}^{2}+p_{4}^{2}$. Then $g$ is a strictly
positive polynomial in the boundary of $\Sigma_{4,4}$. The decomposition of
$g$ as a sum of squares is unique up to orthogonal transformations (see [3,
Section 3.13]).

We add to this example four new polynomials in a ring with one more variable.

######  Example 2.2.

In $\mathbb{R}[x_{1},\dots,x_{5}]$, let $p_{1},p_{2},p_{3},p_{4}$ be as in
Example 2.1, and set

$p_{5}=x_{1}x_{5},\quad p_{6}=x_{2}x_{5},\quad p_{7}=x_{3}x_{5},\quad
p_{8}=x_{4}x_{5}.$

Let
$g=p_{1}^{2}+p_{2}^{2}+p_{3}^{2}+p_{4}^{2}+p_{5}^{2}+p_{6}^{2}+p_{7}^{2}+p_{8}^{2}$.
Then $g\in\Sigma_{5,4}$ is the sum of 8 squares and cannot be decomposed as
the sum of 7 squares.

######  Remark 2.3.

The intuitive idea behind this example is that since the monomial $x_{5}^{4}$
is not in $g$, the monomials $x_{i}^{2}x_{5}^{2}$, $1\leq i\leq 4$, in $g$ can
only be obtained from a product $(x_{i}x_{5})^{2}$, hence
$x_{1}x_{5},\dots,x_{4}x_{5}$ can be thought as _new_ variables.

We will prove the claim in the example by brute force, that is, we will
suppose that there is a decomposition of $g$ as a sum of 7 squares and prove
that the resulting equations on the coefficients have no real solution. To
reduce the redundancy and simplify the computations, we show first that the
polynomials in the decomposition can be assumed to be in triangular shape.

######  Lemma 2.4.

Let $\\{p_{1},\dots,p_{s}\\}\subset k[x_{1},\dots,x_{n}]$ be a set of $s$
linearly independent homogeneous polynomials of the same degree $d$ and let
$g$ be a sum of squares, $g=\sum_{i=1}^{t}q_{i}^{2}$, where $q_{i}\in\langle
p_{1},\dots,p_{s}\rangle_{\mathbb{R}}$, $1\leq i\leq t$ (each $q_{i}$ is a
real linear combination of $p_{1},\dots,p_{s}$). Then for some
$t^{\prime}\leq\min(t,s)$ there exists a decomposition of $g$ as a sum of
$t^{\prime}$ squares $g=\sum_{i=1}^{t^{\prime}},\tilde{q}_{i}^{2}$ where the
polynomials $\tilde{q}_{i}\in H_{n,d}$ are in triangular shape with respect to
$p_{1},\dots,p_{s}$. That is, for any $1\leq i\leq t^{\prime}$,
$\tilde{q}_{i}\in\langle p_{i},\dots,p_{s}\rangle_{\mathbb{R}}$.

###### Proof.

Let $g=q_{1}^{2}+\dots+q_{t}^{2}$, with $q_{i}=\sum_{j=1}^{s}a_{ij}p_{j}$,
$a_{ij}\in\mathbb{R}$, $1\leq i\leq t$, $1\leq j\leq s$. Taking
$v=\begin{pmatrix}p_{1}&p_{2}&\dots&p_{s}\end{pmatrix}^{T}$, we get

$g=v^{T}A^{T}Av,$

where $A\in\mathbb{R}^{t\times s}$ is the matrix with entries $a_{ij}$.

The matrix $M=A^{T}A\in\mathbb{R}^{s\times s}$ is a positive semidefinite
matrix of rank at most $\min(t,s)$. Let $t^{\prime}$ be the rank of $M$. Any
such matrix can be decomposed as $M=B^{T}B$, where
$B\in\mathbb{R}^{t^{\prime}\times s}$. Now let $B=QR$ be a QR decomposition of
$B$, with $Q\in\mathbb{R}^{t^{\prime}\times t^{\prime}}$ orthogonal and
$R\in\mathbb{R}^{t^{\prime}\times s}$ upper triangular. Then $B^{T}B=R^{T}R$
and the formula $g=v^{T}R^{T}Rv$ gives the decomposition of $g$ in triangular
shape, taking $\tilde{q}_{i}$ as the $i$th element of $Rv$. ∎

######  Proposition 2.5.

The polynomial $g$ in Example 2.2 is a sum of 8 squares and cannot be
decomposed as the sum of 7 squares.

###### Proof.

See [5] for the computations in Maple [7]. We first prove that any polynomial
in a SOS decomposition of $g$ is a linear combination of
$\\{p_{1},\dots,p_{8}\\}$. We look for non-zero linear forms
$\ell\in\Sigma_{5,4}^{*}$ that vanish in $g$. By [1, Lemma 2.6] (see also [6,
Proposition 5.2]), $\ell$ must satisfy $\ell(p_{i}q)=0$ for all $p_{i}$,
$1\leq i\leq 8$, and all $q\in H_{5,2}$. We associate to each linear form
$\ell$ the bilinear form $Q_{\ell}(p,q)=\ell(pq)$ and consider the set of
bilinear forms $\\{Q_{\ell}:H_{5,2}\times
H_{5,2}\rightarrow\mathbb{R}\mid\ell\in H_{5,4}^{*},\ell(p_{i}q)=0\text{ for
all $p_{i}$, $1\leq i\leq 8$, and all $q\in H_{5,2}$}\\}$. For the monomial
base

$\mathcal{M}=\\{x_{1}^{2},x_{1}x_{2},x_{1}x_{3},x_{1}x_{4},x_{2}^{2},x_{2}x_{3},x_{2}x_{4},x_{3}^{2},x_{3}x_{4},x_{4}^{2},x_{1}x_{5},x_{2}x_{5},x_{3}x_{5},x_{4}x_{5},x_{5}^{2}\\}$

we compute the space of matrices $\mathsfit{Q}$ corresponding to those
bilinear forms . We obtain a 2-dimensional space
$E=\\{\mathsfit{Q}(t_{1},t_{2}):t_{1},t_{2}\in\mathbb{R}\\}$, where
$\mathsfit{Q}(t_{1},t_{2})=\left(\begin{array}[]{c|c|c}t_{1}A&&\\\
\hline\cr&\bm{0}_{4\times 4}&\\\
\hline\cr&&t_{2}\end{array}\right)\in\mathbb{R}^{15\times 15}$ with

$A=\begin{pmatrix}6&-1&-1&1&6&-1&1&6&1&6\\\ -1&6&-1&1&-1&-1&1&-1&1&-1\\\
-1&-1&6&1&-1&-1&1&-1&1&-1\\\ 1&1&1&6&1&1&-1&1&-1&1\\\
6&-1&-1&1&6&-1&1&6&1&6\\\ -1&-1&-1&1&-1&6&1&-1&1&-1\\\
1&1&1&-1&1&1&6&1&-1&1\\\ 6&-1&-1&1&6&-1&1&6&1&6\\\ 1&1&1&-1&1&1&-1&1&6&1\\\
6&-1&-1&1&6&-1&1&6&1&6\end{pmatrix}.$

A linear form $\ell\in H_{5,4}^{*}$ is in $\ell\in\Sigma_{5,4}^{*}$ iff the
bilinear form $Q_{\ell}$ is positive semidefinite. The matrix
$\mathsfit{Q}(t_{1},t_{2})$ is a block matrix, with a block depending on
$t_{1}$ and a block depending on $t_{2}$. The matrix $A$ is positive
semidefinite, so for any positive numbers $t_{1}$ and $t_{2}$ the matrix
$\mathsfit{Q}(t_{1},t_{2})$ is positive semidefinite. Setting
$\mathsfit{Q}=\mathsfit{Q}(1,1)$, the resulting matrix has kernel of dimension
8:

$W=\langle p_{1},\dots,p_{8}\rangle_{\mathbb{R}},$

which proves our first claim, because any polynomial in a SOS decomposition of
$g$ must be in the kernel of $\mathsfit{Q}$ (see Section 1.1.3).

Now we prove that there is no decomposition of $g$ as a sum of 7 squares. By
the first part and Lemma 2.4, we can assume that the polynomials in the
decomposition are in triangular shape with respect to $p_{1},\dots,p_{8}$.
That is, we can assume

$\begin{array}[]{r*{18}{l}}f_{1}&=u_{11}p_{1}&+&u_{12}p_{2}&+&u_{13}p_{3}&+&u_{14}p_{4}&+&u_{15}p_{5}&+&u_{16}p_{6}&+&u_{17}p_{7}&+&u_{18}p_{8},\\\
f_{2}&=&&u_{22}p_{2}&+&u_{23}p_{3}&+&u_{24}p_{4}&+&u_{25}p_{5}&+&u_{26}p_{6}&+&u_{27}p_{7}&+&u_{28}p_{8},\\\
f_{3}&=&&&&u_{33}p_{3}&+&u_{34}p_{4}&+&u_{35}p_{5}&+&u_{36}p_{6}&+&u_{37}p_{7}&+&u_{38}p_{8},\\\
f_{4}&=&&&&&&u_{44}p_{4}&+&u_{45}p_{5}&+&u_{46}p_{6}&+&u_{47}p_{7}&+&u_{48}p_{8},\\\
f_{5}&=&&&&&&&&u_{55}p_{5}&+&u_{56}p_{6}&+&u_{57}p_{7}&+&u_{58}p_{8},\\\
f_{6}&=&&&&&&&&&&u_{66}p_{6}&+&u_{67}p_{7}&+&u_{68}p_{8},\\\
f_{7}&=&&&&&&&&&&&&u_{77}p_{7}&+&u_{78}p_{8}.\end{array}$

The equation $g=f_{1}^{2}+\dots+f_{7}^{2}$ defines a system of quadratic
equations in the coefficients $u_{ij}$. To solve this system, we compute a
Groebner basis of the ideal defined by the equations using the degree reverse
lexicographical ordering. We obtain the basis $\\{1\\}$, which proves that
there is no solution to the system of equations. That is, $g$ cannot be
decomposed as a sum of 7 squares. ∎

######  Remark 2.6.

In the first part of the above proof, we have shown that the polynomial $g$ in
Example 2.2 is a sum of at most 8 linearly independent squares. This means
that the Gram spectrahedron of $g$ contains points of rank at most $8$. In the
second part we have shown that the Gram spectrahedron of $g$ contains no point
of rank $7$. Both results combined imply that the Gram spectrahedron consists
of only one point (or otherwise there would be points of different rank). That
is, the decomposition of $g$ given in Example 2.2 is the unique (up to
orthogonal equivalence) decomposition of $g$ as a sum of squares.

This can also be verified computationally adding an eighth polynomial
$f_{8}=u_{88}x_{4}x_{5}$ in the above proof and computing the Groebner basis
of the resulting system of equations (see the auxiliary code [5] for this
computation).

######  Remark 2.7.

If we follow this strategy to construct examples for other values of $n$, we
would start with a sequence of parameters of degree 2 in $n-1$ variables,
hence consisting of $n-1$ polynomials, and add $n-1$ more polynomials
$x_{1}x_{n},x_{2}x_{n},\dots,x_{n-1}x_{n}$. The total number of polynomials is
then $2(n-1)$. If it also holds that the resulting polynomial cannot be
decomposed as a sum of less polynomials, this bound would improve the bound
given in [8] only in the case $(n,d)=(5,4)$, which is the case studied in this
paper, so we don’t attempt to extend our construction to larger values of $n$.

## References

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  * [5] Santiago Laplagne, _Auxiliary code for this paper_ , https://bitbucket.org/slaplagne/pythagoras-numbers/, 2023.
  * [6] Santiago Laplagne and Marcelo Valdettaro, _Strictly positive polynomials in the boundary of the sos cone_ , 2020, arXiv:2012.05951.
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