# Master of Disaster: A Disaster-Related Event Monitoring System From News
Streams

Junbo Huang
Department of Computer Science
University of Hamburg
Vogt-Kölln-Straße 30, 22527 Hamburg, Germany
<EMAIL_ADDRESS>
Ricardo Usbeck
Head of the AI and Explainability Group
Leuphana University Lüneburg
Universitätsallee 1, 21335 Lüneburg, Germany
<EMAIL_ADDRESS>

###### Abstract

The need for a disaster-related event monitoring system has arisen due to the
societal and economic impact caused by the increasing number of severe
disaster events. An event monitoring system should be able to extract event-
related information from texts, and discriminates event instances. We
demonstrate our open-source event monitoring system, namely, Master of
Disaster (MoD), which receives news streams, extracts event information, links
extracted information to a knowledge graph (KG), in this case Wikidata, and
discriminates event instances visually. The goal of event visualization is to
group event mentions referring to the same real-world event instance so that
event instance discrimination can be achieved by visual screening.

_K_ eywords Event Extraction $\cdot$ Knowledge Graph $\cdot$ Event Instance
Discrimination

##  1 Introduction

The increased frequency and magnitude of disasters in recent years have led to
significant human, environmental, and economic losses. In response to this,
disaster-related event monitoring systems, such as GDELT (Leetaru and Schrodt,
2013) and GDACS (De Groeve et al., 2006), have emerged as critical tools to
(1) monitor events and (2) provide timely overviews for an early disaster
response planning. Such large-scale systems combine manual and automated
approaches in retrieving event data from sensory and raw text data. Advances
in machine learning have significantly boosted predictive models’ ability to
extract accurate information from raw texts in the above-mentioned monitoring
systems. However, a major challenge on discriminating event instances remains
unsolved. In other words, how do we know if multiple news articles are
referring to the same real-world event instance?

Event instance discrimination is a less research area. Existing work mainly
relies on entity matching based on factors such as location and time (Rollo
and Po, 2020; Zavarella et al., 2020; Ai et al., 2021), or linking event
mentions to entries in a knowledge graph (KG) like
Wikidata111https://www.wikidata.org/ (Yu et al., 2023). However, heuristic-
driven methods often suffer from poor generalization, while KG-driven
approaches are limited by the inability to represent events absent from the
KG. In this word, we strive for an event monitoring system that (1) receives
news stream; (2) extracts event information; (3)links extracted information to
a Knowledge Graph (KG), in this case, Wikidata and (4) discriminates event
instances visually.

To achieve this goal, we suggest that a half-automated, human-in-the-loop
event monitoring system can leverage predictive models to provide reliable
information for humans. Such information includes users, geospatial and
temporal information of events as well as an indication on whether two or more
event mentions are referring to the same real-world event instance.

In this work, we present Master of Disaster (MoD)222The online demo can be
found here https://hitec.demo.skynet.CoyPu.org/. Please ignore tabs other than
Event Extraction and Event Visualization. The source code for the GUI can be
found here https://github.com/semantic-systems/SEMS-tool-suite/. The source
code for the event extractor can be found here https://github.com/semantic-
systems/CoyPu-EventExtraction, a half-automated, human-in-the-loop disaster-
related event monitoring system with three key features. First, MoD receives
daily news streams from GDELT333https://www.gdeltproject.org/ with a keyword
search. Second, MoD extracts event-related information with a transformers-
based event extractor, whose output is linked to Wikidata and the CoyPu
ontology444https://schema.coypu.org/global/. Third, MoD provides a
visualization of events to represent the similarity of different news in the
embedding space. To compensate for the error produced by the event extractor,
a visualization from an unsupervised clustering algorithm is provided as a
reference for the human evaluation of event distributions.

MoD aims to provide an overview of daily events extracted from the user-
provided keywords. Furthermore, it should be mentioned that the semantic
output (linking to Wikidata) helps leverage downstream tasks such as knowledge
validation and common sense reasoning.

##  2 Master of Disaster (MoD)

MoD is an open-sourced application with four components: a data preprocessor,
an event extractor, an event visualizer, and a gradio-based
GUI555https://gradio.app/. The architecture of MoD is illustrated in Figure 1.

Figure 1: Overview over MoD’s Architecture.

###  2.1 Data preprocessor

First, the data preprocessor queries a list of articles containing the user-
provided keywords with the GDELT API666Note that the GDELT API restricts the
maximal amount of returned articles to 250.. Furthermore, it applies a
filtering method to select articles that are written in English. This is
because the event extractor can only receive English inputs, currently. With
the filtered articles, the data preprocessor takes the title of each article
to represent an event. Additionally, all artifacts (such as white spaces
around punctuation marks) are removed from the titles to retrieve the final
representation of events in the textual space.

###  2.2 Event Extractor

The event extractor consists of three components, an event type detector, an
entity linker and an RDF graph generator777Resource Description Framework
(RDF) is a standard model for data interchange on the Web.. The event type
detector is a pre-trained RoBERTa-base (Liu et al., 2019) model fine-tuned on
the TREC-IS dataset (McCreadie et al., 2019). The model is optimized using the
supervised contrastive learning (Khosla et al., 2020) loss. In the original
TREC-IS benchmark, the task was to predict information types from raw texts,
such as Request-GoodsServices or Report-EmergingThreats. However, we modified
the original task schema to predict the event type888In total, there are 9
event types: tropical storm, flood, shooting, covid, earthquake, hostage,
fire, wildfire and explosion (such as Earthquake or Shooting). Note, the event
type was considered a meta-information in the original dataset. Also, the
dataset did not include an out-of-scope (oos) class describing non-events.
Thus, we included data from the TweetEval dataset (Barbieri et al., 2020) to
represent an oos class. It is worth noting that TweetEval included 7 subtasks,
such as Emotion Recognition, Irony Detection and Hate Speech Detection.
Sentences in TweetEval are all user-generated tweets, which contain a variety
of topics, language styles and structures than traditional news articles. By
integrating such samples in the training data, the model learned a more
concentrated feature representation for the given event classes.

Next, we employed an entity linker to extract event-related entities. MoD
presently considers two relations, namely hasLocality and hasImpactOn,
following the CoyPu Event
ontology999https://schema.CoyPu.org/global/2.3#Event. Here, we leveraged a
local entity linker, namely
BLINK101010https://github.com/facebookresearch/BLINK (Wu et al., 2020), a
BERT-based entity linking model which links entities to Wikipedia. However,
since the CoyPu ontology is based on Wikidata as background KG, the RDF graph
generator maps from Wikipedia to Wikidata and outputs an event graph in JSON-
LD (see Appendix A).

###  2.3 Event Visualizer

The event visualizer (1) receives a stream of filtered articles from the data
preprocessor, (2) followed by the event type detector to generate sentence
embeddings from the last hidden layer and make predictions on event types, and
(3) visualizes the sentence embeddings in a two-dimensional coordinate. The
goal of event visualizer is to group event mentions referring to the same
real-world event instance. We applied a dimensionality reduction technique
based on Principal Components Analysis (PCA). Two resulting two-dimensional
visualizations are presented in MoD, as shown in Figure 2.

Figure 2: Event Visualization with respect to the classification (left) and
clustering (right) Result. The search keyword is "Hamburg", and the date is
"11. March 2023"

The first one illustrates the event type prediction by the event type detector
(left-hand side in Figure 2) and the second visualization (right-hand side in
Figure 2) introduces the prediction from a clustering algorithm. We used
DBSCAN (Ester et al., 1996) as the clustering algorithm because DBSCAN is a
density-based algorithm, where the number of clusters is not pre-defined.
Therefore it can group news into an arbitrary number of event instances. From
the visualization shown in Figure 2, we have observed that DBSCAN considers
local structures (measured by the distance between sentence representations)
of sentence embedding in the PCA-reduced two-dimensional space. This provides
insights for human evaluators to have a direct access to event distribution,
which is important to discriminate between event instances.

The final component in MoD is the gradio-based interactive GUI which is a
centralized, injective component that makes API calls to all services used in
MoD. These services include an event extractor and an event visualizer.

##  3 Evaluation

To ensure that MoD can be practically deployed as a half-automated disaster-
related event monitoring system, we conducted a survey with seven NLP
researchers, among which five of the participants are frequent news readers.
The only demographic that we controlled for is the expertise in NLP. We
observed that six participants have expertise in knowledge graphs and three
participants have expertise in event extraction. The main investigation in the
survey is the usability of MoD, which consists of the following questions on a
5-point Likert scale.

  1. 1.

How easy is it to use the Event Extraction tab? (1: very difficult; 5: very
easy)

  2. 2.

How accurate is the information provided by the event extractor? (1: very
inaccurate; 5: very accurate)

  3. 3.

How easy is it to use the Event Visualization tab? (1: very difficult; 5: very
easy)

  4. 4.

Is the overview of daily news by the Event Visualizer informative? (1: very
non-informative; 5: very informative)

  5. 5.

Will you use Event Visualizer for fast access to the news? (1: no; 5: yes)

As a result, we received 4.14 points on average on both the first and the
third question (level of easiness to use the system); 3 points on average on
the third question (accuracy by the event extractor); 3.57 points on average
on the fourth question (informativeness of the event visualizer) and 3.14
point on average on the last question (practicality). Albeit with a small set
of participants, we observed that the majority of the participants agreed that
MoD is easy to use and the visualization is informative.

##  4 Conclusion and future work

In this paper, we presented MoD, a disaster-related event monitoring system.
Moreover, we explained the architecture of MoD and the functionality of each
component. The goal of MoD is to provide an event monitoring system that (1)
reveives news stream; (2) extracts event information; (3) links the extracted
information to a KG and (4) discriminates event instances mentioned in the
news. In future work, we aim to provide a temporal analysis of events and
increase the performance of the event extractor.

## Ethics Statement

To our knowledge, this work does not concern any substantial ethical issue.
Corpora used in this work are preprocessed by masking all user mentions and
links. Of course, the application of classification algorithms could always
play a role in Dual-Use scenarios. However, we consider our work as not risk-
increasing.

## Acknowledgements

The authors acknowledge the financial support by the Federal Ministry for
Economic Affairs and Energy of Germany in the project CoyPu (project number
01MK21007[A-L]). HITeC is “G” This work was also supported by the House of
Computing and Data Science (HCDS) of the Hamburg University within the Cross-
Disciplinary Lab programme, and by the Ministry of Research and Education
within the project ‘RESCUE-MATE: Dynamische Lageerstellung und Unterstützung
für Rettungskräfte in komplexen Krisensituationen mittels Datenfusion und
intelligenten Drohnenschwärmen’ (FKZ 13N16844).

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##  Appendix A Example output graph of the event extractor

⬇

{

@id: "https://data.CoyPu.org/event/mod/d868a8be-c49e-48b8-a3a5-5b4b12d7d97f",

@type: [

0: "https://schema.CoyPu.org/global#Event" ,

1: "http://www.wikidata.org/entity/Q2252077"

],

http://www.w3.org/2000/01/rdf-schema#comment: [

0: {

@value: "Hamburg shooting : Multiple dead after attack at

Jehovah Witness church in Germany"

}

],

https://schema.CoyPu.org/global#hasImpactOn: [

0: {

@id: "http://www.wikidata.org/entity/Q35269"

}

],

https://schema.CoyPu.org/global#hasLocality: [

0: {

@id: "http://www.wikidata.org/entity/Q1055"

} ,

1: {

@id: "http://www.wikidata.org/entity/Q183"

}

],

https://schema.CoyPu.org/global#hasPublisher: [

0: {

@value: "HiTec"

}

],

https://schema.CoyPu.org/global#hasTimestamp: [

0: {

@value: "15_03_2023_17_57_56"

}

]

}


#  Fluidic FlowBots: Intelligence embodied in the characteristics of
recirculating fluid flow
††thanks:  M. Gepner, J. Mack, F. Giorgio-Serchi, and A.A. Stokes
$($adam.stokes@ed.ac.uk$)$ are with The School of Engineering and The
Edinburgh Centre for Robotics at The University of Edinburgh, The King’s
Buildings, West Mains Road, Edinburgh, Scotland, EH9 3FF.
The authors gratefully acknowledge the supported of the EPSRC CDT-RAS
EP/S023208/1
For the purpose of open access, the author has applied a Creative Commons
Attribution (CC BY) licence to any Author Accepted Manuscript version arising
from this submission. *Address all correspondence to this author.

Maks Gepner  Jonah Mack  Francesco Giorgio-Serchi, Member IEEE and Adam A.
Stokes∗, Member IEEE

###### Abstract

The one-to-one mapping of control inputs to actuator outputs results in
elaborate routing architectures that limit how complex fluidic soft robot
behaviours can currently become. Embodied intelligence can be used as a tool
to counteract this phenomenon. Control functionality can be embedded directly
into actuators by leveraging the characteristics of fluid flow phenomena.
Whilst prior soft robotics work has focused exclusively on actuators operating
in a state of transient/no flow (constant pressure), or pulsatile/alternating
flow, our work begins to explore the possibilities granted by operating in the
closed-loop flow recirculation regime. Here we introduce the concept of
FlowBots: soft robots that utilise the characteristics of continuous fluid
flow to enable the embodiment of complex control functionality directly into
the structure of the robot. FlowBots have robust, integrated, no-moving-part
control systems, and these architectures enable: monolithic additive
manufacturing methods, rapid prototyping, greater sustainability, and an
expansive range of applications. Based on three FlowBot examples: a
bidirectional actuator, a gripper, and a quadruped swimmer - we demonstrate
how the characteristics of flow recirculation contribute to simplifications in
fluidic analogue control architectures. We conclude by outlining our design
and rapid prototyping methodology to empower others in the field to explore
this new, emerging design field, and design their own FlowBots.

##  I Introduction

###  I-A Embodied intelligence expands the capabilities of fluidic soft
robots.

The field of robotics has seen a plethora of developments in using fluid
pressure (pneumatic and hydraulic) to actuate compliant, soft-bodied robots
[1]; these soft-robots exhibit superior capabilities over their rigid
counterparts in use cases requiring high dexterity and safety during operation
with humans [2, 3]. Replacing electronic controls with fluidics has also
opened up the possibility of deploying soft robots in extreme environments,
thanks to the inherent high power density, and temperature and radiation
resistance of fluidic systems [4, 5, 6].

There have been a range of notable advancements in recent years for
controlling fluidic soft robots using hardware that mimics digital electronic
devices and architectures [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22]. Despite these advancements, scaling fluidic control system to
higher degrees of complexity remains difficult; a large number of pressure
control inputs is required, and routing control signals becomes an issue [4,
23].

Attempts at overcoming this “routing problem” have included fluidic
demultiplexer chips [15], oscillators [13], fluidic matrix circuits [23], as
well as analogue devices that forego the need for discretisation [14, 24]. An
alternative approach involves embodied intelligence, whereby control features
can be embedded directly into the robot. By exploiting the mechanics of the
interaction of the robot with its physical operating environment as a form of
sensory feedback, the need for external control inputs can be reduced [25, 26,
27, 28].

###  I-B Internal flow characteristics can embed functionality into the
structure of the actuator.

In most reported-cases, and for each of the actuators cited by Section I-A,
the pressure is adjusted by pumping set quantities of fluid between the
individual chambers of the actuator; this transition is carried out in a one-
off, pulsatile, or cyclic manner. Due to the mechanical coupling between the
fluid medium and the vessels/channels containing it, the behaviour of a
fluidic actuator can be directly influenced by controlling the pressure of the
fluid inside of it. During the pumping phase, the flow rate of the fluid is
determined by its interaction with the flow channels; this mechanical coupling
influences the rate of change of the pressure distribution inside the
actuator, and, therefore, the deformation of the actuator. If the
characteristics of the flow can be exploited, they can be used to control the
actuator.

Viscous energy losses lead to a decrease in the pressure of the flowing fluid.
Previous work in the field has explored exploiting this phenomenon for control
purposes [29, 30, 31, 32, 33]. In their recent work [34], Matia et al. used
the resultant pressure asymmetries to control the gait of a walking robot.
Matia et al. accurately modelled the relationship between the timing of
viscous phenomena and the inertial coupling between the fluid and the
actuator; therefore, they demonstrated how controlling the timing of transient
fluid flows can serve as a form of embodied intelligence.

Whilst there as several other notable examples of previous fluidic soft robots
that used “pulsatile”, or “cyclic” modes of flow (such as [35, 8, 36, 37, 19,
38]), a condition that has not yet been explored and well-characterised in the
context of soft fluidic robots is a state of continuous flow recirculation.
Figure 1 illustrates the difference between soft fluidic robots based on the
“steady state”, “pulsatile/cyclic”, and “recirculating” modes of flow, as
defined in this paper. Fluid can be recirculated in a system for example to
achieve thermal regulation, as in the work of Cacucciolo et al. [39], and
others [40, 41]. Liao et al. [41] showed that the pressure of this
recirculating fluid can also be used to inflate the cavities of an actuator,
achieving deformation whilst simultaneously taking advantage of the thermal
regulation characteristic (which they used to modify the stiffness of the
actuator).

In this paper, we delve deeper into the behaviour of soft robots operating
under the closed-loop fluid recirculatory condition, describing the
opportunities granted by it in the way of embodied intelligence, as well as
some of the associated design challenges. We aim to empower designers to build
a whole new class of fluidic robots, which we term “FlowBots” (as further
described by Section I-D).

Figure 1: Soft fluidic robots utilising a) ”steady state” b)
”pulsatile/cyclic” c) ”recirculating” flow, as defined in this paper.

###  I-C Intelligence embodied in fluid flow characteristics: from shark
intestines and Tesla-valves to soft robots that can be printed as one part.

The state of “recirculation” is characterised by the driving fluid, taken from
a reservoir, being pumped, through the channels in the actuator channels, and
back to the reservoir, as illustrated on Figure 1. To use an electrical
analogy, one can think of the actuator as a source of impedance in a closed-
loop fluidic network. The ”current” (the driving fluid) flows from a ”high
potential” (outlet of the pump/pressure source), through the ”impedance”
(internal channels in the actuator), and back to a low potential (the fluid
reservoir from which the fluid was originally pumped). Energy is lost via
viscous friction, which constitutes the resistive component of the fluidic
impedance, and the drop in fluid pressure that results is analogous to a drop
in electrical potential.

There are also other fluid flow phenomena that can contribute to a change in
the pressure distribution in the system. Some of these phenomena deal with the
inertial component of the fluidic impedance, which is analogous to electrical
inductance. A simple example is the Tesla-valve [42], where the geometry of
the device results in pronounced impedance in one flow direction compared to
the other. As visualised by Figure 2(a,b), the added secondary flow ”loops”
impede the main flow when they collide with it, as a result of the inertia
they carry. In nature, a similar geometry can be found in the intestinal
tracts of certain species of sharks, skates, and rays. Leigh et al. [43]
demonstrate that this geometry slows the flow in one direction due to the
added fluidic impedance; they hypothesise that this geometry helps the animals
slow the rate of food transit (thereby improving nutrient absorption in the
short intestinal tract), as well as reducing the energy cost of digestion due
to promoting flow in only one direction. The advantage of this type of
geometry is that it avoids moving components (such as the flaps found in heart
valves), which would be prone to blockage.

Figure 2: Inspiration for exploiting fluid flow phenomena as a form of
embodied intelligence can be found in nature [43]. The geometry of the shark
intestine (c) resembles that of a Tesla-valve (a, b). Flow circulation induced
by the bellowed geometry results in a diode-like behaviour, making it harder
for fluid to flow in one direction (left-to-right as pictured on the figure)
than the other (right-to-left). This type of fluidic diode has no moving
components; the added functionality is embodied into the structure of the
device.

###  I-D FlowBots: a new class of soft robots with intelligence embedded in
the characteristics of recirculating fluid flow.

This paper focuses on embedding additional functionality directly into the
bodies of soft robots, by taking advantage of the properties of recirculating
fluid flow. Here, we demonstrate that new types of soft robots can be
manufactured as a single part, with embedded control, and actuation. This mode
of manufacture enables rapid prototyping, increases robustness, reduces waste,
and minimises the risk of environmental pollution (opening up opportunities
for new applications in e.g. wildlife biodiversity monitoring). We term such
robots “FlowBots”.

FlowBots are designed to operate under the state of recirculating flow, as
defined by Section I-B; this characteristic differentiates them from previous
soft robots controlled and actuated using fluids. Here, fluid flows into an
inlet port and flows out through an outlet port. A single actuator can have
multiple inlets and outlets, as well as additional control, or sensing ports
if needed. FlowBots are capable of complex behaviours with only a single
pressure source, due to their embodied fluidic intelligence.

The focus of this paper is to demonstrate the principles of operation of
FlowBots based on three simple examples: a bidirectional actuator, a gripper,
and a quadruped swimmer robot. Each example builds upon the operating
principles of the previous one. The bidirectional actuator demonstrates how
pressure asymmetries arising due to viscous losses can be used for analogue
control using localised flow channel constrictions. The gripper shows how flow
recirculation can simplify control architectures of multi-actuator systems due
to the ability of switching between connection in parallel, and in series. The
swimmer FlowBot builds upon the control architecture used for the gripper, and
serves as a demonstration of how the embodiment of intelligence in fluid flow
characteristics allows to create robots capable of complex behaviours, yet
manufactured as single, robust components, without any moving parts. Section
II describes some of our experimental work in evaluating the performance of
FlowBots with air and water as the operating fluid. We conclude with a
discussion on some of the design challenges associated with building FlowBots,
empowering others in the field to build their own.

##  II Results

###  II-A Localised pressure asymmetries arising due to viscous losses in
fluid flow can be used for analogue, bidirectional control.

Figure 3 illustrates the operation of the first FlowBot described in this
paper – a bidirectional actuator. The left and right pressure chambers of the
actuator are connected at the tip to allow for recirculation of flow from and
back to the pressure source in a closed loop. Energy is lost due to the
friction between the flowing fluid and the walls of the channel, and the
pressure of the fluid progressively decreases as it advances along the flow
path constituted by the internal channels inside the actuator. By
concentrating these viscous energy losses at the tip of the actuator, a
pressure distribution is obtained that is symmetric along the longitudinal
centreline of the actuator; the chamber in one half has high pressure, and the
other half has low pressure. This pressure asymmetry results in deformation in
one direction. When the flow direction is reversed, the asymmetry flips, and
the actuator deforms in the opposite direction.

If the location of the constriction was moved further upstream, or downstream
of the tip, the deformation characteristic of the actuator would be
permanently skewed to one side. Having the actuator deform differently in both
directions at the same pressure difference could be desirable in certain
applications. In this paper, all the actuators are symmetric, but we utilise
external variable constrictions to regulate the direction of flow, and the
pressure difference across the actuator. These constrictions can be
implemented using additional valves, by manually blocking/kinking the outlet
vents/tubes, or by using soft buttons (see the supplementary video). Figure 3
shows how we were able to use two such constrictions to achieve analogue
control of the actuator in two directions, by regulating the pressure across
it over a continuous spectrum. This analogue mapping can also be achieved by
adjusting the direction of flow, as well as the flow rate of the supply pump
(hydraulic case), or compressed air supply pressure (pneumatic case).

Figure 3: The principle of operation of the bidirectional actuator. By varying
the outlet vent constrictions, the direction, and magnitude of the flow, as
well as the pressure difference across the actuator can be controlled with an
analogue characteristic. Reversing the flow direction causes the actuator to
deform symmetrically in the other direction due to the mirroring of the
pressure asymmetry. The percentage value of the constriction is an
approximation; we estimated it indirectly by measuring the pressure difference
across the actuator using two pressure gauge probes.

###  II-B FlowBots have simpler control architectures due to being able to
operate in parallel, as well as in series.

Section II-A demonstrated how to control a single bidirectional actuator, two
controls (vent outlet constrictions) are used. To fully control two such
actuators, four controls would normally be needed. By exploiting the
characteristics of recirculating fluid flow, we were able to reduce this fan-
in to three. The simplification comes from the fact that under the state of
recirculating flow, the actuators can operate not only in a parallel, but also
a series connection. The ability to switch between the two configurations
allows to simplify the control architecture, and opens up the design space for
constructing FlowBots consisting of several actuators, that are capable of
more complex behaviours.

Figure 4 illustrates the operation of the gripper that we have built using two
bidirectional actuators described in Section II-A. The gripper uses only three
control inputs rather than four, yet each of its fingers can still be
controlled fully independently. The last two configurations are made possible
by the fact that if the middle port is blocked, the two actuators become
connected in series rather than in parallel. This capability would not be
possible without using recirculating fluid flow in the system.

Figure 4: The principle of operation of a gripper consisting of two
bidirectional actuators. By switching between a parallel ((a) to (g)) and
series ((h) and (i)) connection of the actuators, we were able to achieve
fully independent analogue control of each of the fingers using only three
control inputs rather than four. Note that in the series configurations the
actuators deform slightly less, due to the pressure difference across each one
being effectively half of that in the parallel configuration.

###  II-C Complex FlowBots can be additively manufactured as one part, with no
moving components.

The control characteristics of FlowBots are embodied in the interaction of the
fluid flow, and the structure of the robot structure. No added parts, or
moving components are required, and, as a result, entire robots can be
additively manufactured as a single part, in one print job. Figure 5 depicts
the operation of a quadruped swimming robot that we have built based on the
control architecture used for the gripper design described in Section II-B. As
in the case of the gripper, we can fully control each pair of limbs for this
FlowBot using only three control inputs rather than four, thanks to the
ability to switch between parallel and series connection of the actuators.
Despite the added complexity of the design, the entire 3D stack of the control
architecture comes packaged into the body of the robot, and has been
additively manufactured as a single component. After the print job is over,
the robot is ready to use. Embodying intelligence in this way makes it easy to
prototype FlowBots with ever more complex 3D designs.

Figure 5: Operation of a quadruped swimmer FlowBot that builds upon the
control architecture used for the gripper described by Section II-B. Due to
the embodiment of control into the body of the robot, as well as the
characteristics of recirculating flow, FlowBots of increasing complexity can
still be manufactured as single components.

###  II-D FlowBots can operate with water, or air, as the working fluid.

The mechanical properties of the working fluid affect the characteristics of
the recirculating flow. The fluid viscosity influences the flow rate in the
system. The same flow rate with a more viscous fluid, such as water, will
result in larger viscous losses, and therefore larger pressure drops than with
a less viscous fluid, such as air. Nevertheless, for the same magnitude of
pressure difference across an actuator, we expect the deformation to be
identical, as the deformation of the bellows of the actuator is driven by the
static component of the pressure. To validate this assumption, we conducted an
experiment where we measured the deformation (curvature) of the bidirectional
actuator described in Section II-A with air, and water, as the working fluid.
Figure 6 shows the results of the experiment; these results are discussed in
Section III-A.

To account for the difference in the viscosity of the fluids, we used the
pressure difference between the inlets of the actuator as the independent
variable in the experiment. This approach allowed us to validate whether the
actuator deforms the same amount with the same pressure difference,
independent of whether air, or water, was used as the working fluid. We
examined a range of pressures from 1.25 bar to 2.5 bar (in intervals of 0.25
bar), testing in both flow directions with 3 measurement repetitions per data
point. We also repeated the measurements with another identical actuator
specimen. Due to the symmetry resulting from placing the channel constriction
at the tip of the actuator, we expect it to deform the same amount in both
directions, as explained in Section II-A. For more details about the
experimental setup, please refer to the supplementary material.

Figure 6: The bidirectional actuator deforms consistently with both
incompressible (water), as well as compressible (air) media as the working
fluid. We attribute the observed variations between the measurements to micro-
errors in the manufacturing process on the low-cost 3D printer that we used.
Note that the error bars represent the deviation in measurements between the 2
actuator specimens that were tested. The error between the 3 measurement
samples gathered during each of the experimental runs was negligible. The
measured fluid flow rates at each pressure are annotated on the figure,
although they were not explicitly needed for this specific analysis.

##  III Discussion

###  III-A FlowBots are compatible with both pneumatic, as well as hydraulic
control.

Figure 6 shows the analogue response of the actuator with pressure with both
air and water as the working fluid. We observe that, the deformation of the
actuator is consistent between the two operating media in both directions
within the examined pressure range. There was a variance in the deformation
between the two directions of up to 11%; the examined specimens deform
slightly more in the counterclockwise direction. Based on our experience
manufacturing and testing different FlowBots, we have observed that this
magnitude of error is common, and is related to micro-errors in the
manufacturing process. One of the main factors at play appears to be bed
levelling, which can cause the printer to extrude a different amount of
material in certain locations. This phenomenon can be more pronounced on
budget 3D printers such as the one used here. Due to the fact that the
actuator samples used in this experiment were printed in the same location and
orientation on the bed in two sequential print runs, it is possible that the
bed was misaligned in one direction, or became misaligned during the print,
and contributed to the error.

The channels in the actuators featured in the experiment are as narrow as 0.6
mm, and variations on the order of hundreds of microns at different locations
along the flow path can disturb the symmetry between the pressure
distributions in both directions; such deviations can contribute to marginal
differences in the deformation in two directions. In our case, these errors
did not significantly affect the robot’s desired functionality, but it might
be a factor to consider for those designing FlowBots with very specific
precision requirements. In these instances, we recommend carrying out adequate
testing to ensure that the desired level of precision has been achieved.

Please refer to the supplementary material for a discussion of our experiences
with mitigating other micro-sources of variance that we have observed when
carrying out our experiments, such as the hygroscopic properties of the
elastic TPU filament that we used to print the robots, temperature variations
in the working fluid reservoir, or small internal, or external leaks in the
membranes of the actuators.

In spite of the uncertainty associated with the detected errors in
measurement, our results demonstrate that FlowBots can operate consistently
using both compressible, as well as incompressible media (air and water) as
the driving fluid. This characteristic of cross-compatibility gives greater
flexibility to designers and makes it possible to deploy FlowBots in a wider
range of operating scenarios.

###  III-B Recirculating flow, and facile 3D Printing enables a wide design-
space and low-barrier to entry for researching and using FlowBots.

This paper demonstrates how designers can come up with entirely new fluidic
soft robot concepts by taking advantage of the characteristics of fluid flow.
Constant recirculation of the working fluid comes with the cost of energy
expenditure, but it allows to simplify control architectures (as demonstrated
in Section II-B), and to rapidly prototype and additively manufacture entirely
new FlowBot designs. For additional design considerations to take into account
when adapting systems to use recirculating flow, please refer to the
supplementary material, where we have discussed these points in further
detail.

Recirculating flow
The exact behaviour of the fluid inside of the internal channels is difficult
to characterise analytically due to the turbulence arising as a result of the
combination of bellow geometries, multi-directional pipe bends, and varying
channel cross-sections. Whilst we do invite others specialising in the domain
to help better characterise these phenomena, we encourage those interested in
coming up with their own FlowBot designs to take advantage of Finite Element
Method (FEM) software. In our design methodology, we have been able to utilise
Computational Fluid Dynamics (CFD) to obtain an approximation of the internal
flow field that was more than adequate enough to facilitate effective design
exploration and validation. By combining CFD with structural simulations using
Finite Element Analysis (FEA) software we were able to simultaneously explore
how modifying parameters in the design affects the fluid flow in the control
system, as well as the mechanical deformation of the actuators.

3D printing
With our chosen method of Fused Deposition Modelling (FDM) 3D printing, we
were able to manufacture all of the FlowBots featured in this paper as one
part, without any supports or additional post-processing needed. Once taken
off the print bed, each FlowBot is ready to use, as it integrates actuation,
control, and even fluidic fittings in its design. In spite of the complex
bellow geometry of its actuators, and the multi-layered internal 3D control
architecture, the swimmer FlowBot described in Section II-C took under 7 hours
to print on a cheap, $200 Artillery Sidewinder X1 printer using Recreus
Filaflex 70A filament (as well as 82A for the specimens in Section II-D). This
design approach is more environmentally sustainable, adds robustness and saves
mass and space due to the reduced need for complex routing/manifolds that come
with a pressurising/valving system.

A design variable that we have not yet explored, but which could help open up
the design space even further is the observation that the fluid being
recirculated throughout the system carries energy. This energy could be
extracted for useful work, for example by integrating rotary tool end
effectors into the robots. We foresee this could be an interesting branch to
explore for those interested in developing novel FlowBot systems, in
particular ones oriented toward more specific industrial applications.

##  IV Conclusion

This paper introduced the concept of FlowBots, a new generation of soft robots
that operate under a state of fluid flow recirculation. FlowBots are
controlled by taking advantage of fluid flow characteristics, allowing for
embodied intelligence. This work demonstrates the operation of three simple
FlowBot systems: a bidirectional actuator, a gripper, and a quadruped swimmer
robot.

By utilising pressure asymmetries arising from viscous losses during fluid
flow, in combination with the ability to operate in series, as well as in
parallel - features only granted by flow recirculation - this work shows that
we have been able to simplify the control architectures required to achieve
fully independent, analogue, and bidirectional control of all of the end
effectors.

Each FlowBot was additively manufactured as a monolithic part with no moving
components, making it low-cost, robust, and easier to dispose of at the end of
its life cycle than multi-part systems utilising different types of materials.
This paper outlines an efficient design methodology that will allow engineers
to rapidly prototype their own FlowBots using FEM methods, and additive
manufacturing.

This paper lays the foundations for a new and wide design space for soft
robotic systems that use recirculating flow.

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# Real analytic functions and monomial curves

János Kollár<EMAIL_ADDRESS>
###### Abstract.

We show that a function is real analytic at the origin iff it is arc-analytic,
has a subanalytic graph, and its restriction to every monomial curve is
analytic. This complements recent results of Kucharz and Kurdyka.

A classical theorem of Hartogs says that a function on ${\mathbb{C}}^{n}$ is
complex analytic iff its restriction to every complex line parallel to a
coordinate axis is complex analytic [Har1906]. For functions on
${\mathbb{R}}^{n}$ the situation is considerably more complicated.

One can check real analyticity on 2-planes. That is, a function
$F(x_{1},\dots,x_{n})$ is real analytic if its restriction to every 2-plane is
real analytic [BS71, BS19]; see [BKK20] for a similar result for 2-spheres.

However, restrictions to lines carry very little information, as shown by the
simple example $xy/(x^{2}+y^{2})$ on ${\mathbb{R}}^{2}$.

Following [Kur88], a function $F(x_{1},\dots,x_{n})$ is called arc-analytic
iff the composite $F\bigl{(}\phi_{1}(t),\dots,\phi_{n}(t)\bigr{)}$ is real
analytic, whenever the $\phi_{i}$ are real analytic. Every real analytic
function is arc-analytic, but there are arc-analytic functions that are not
even continuous; see [BMP91, Kur91, Kur94].

The situation improves if we impose additional restrictions on $F$. [BM90]
shows that if $F$ is arc-analytic and its graph is subanalytic, then $F$
becomes real analytic after a suitable sequence of (local, real analytic)
blow-ups; see Theorem 7 for the statment and [BM88] for the basic theory of
subanalytic sets. It is, however, not clear how to convert this into a
criterion for real analyticity on ${\mathbb{R}}^{n}$ itself.

The recent paper [KK22] proposes an approach that looks more carefully at some
special curves. Their main result says that if $F(x_{1},\dots,x_{n})$ is arc-
analytic and its graph is subanalytic, then $F$ is real analytic at the origin
iff

$F\bigl{(}a_{1}t^{p}+b_{1}t^{q},\dots,a_{n}t^{p}+b_{n}t^{q}\bigr{)}\in{\mathbb{R}}[[t^{p},t^{q}]]\subset{\mathbb{R}}[[t]]$

for every $a_{i},b_{i}\in{\mathbb{R}}$ and $0<p,q\in{\mathbb{N}}$. That is,
$F\bigl{(}a_{1}t^{p}+b_{1}t^{q},\dots,a_{n}t^{p}+b_{n}t^{q}\bigr{)}$ has a
convergent Taylor expansion that involves only the $t$-powers $t^{rp+sq}$ for
$r,s\in{\mathbb{N}}$.

The aim of this paper is to prove the following variant of this result.

######  Theorem 1.

Let $F(x_{1},\dots,x_{n})$ be an arc-analytic function on an open subset
$U\subset{\mathbb{R}}^{n}$ whose graph is subanalytic. Then $F$ is analytic at
the origin iff its restriction to every monomial curve is analytic at the
origin. That is, iff

$None$
$F\bigl{(}t^{m_{1}},\dots,t^{m_{n}}\bigr{)}\in{\mathbb{R}}[[t^{m_{1}},\dots,t^{m_{n}}]]\subset{\mathbb{R}}[[t]]$

for every $0<m_{i}\in{\mathbb{N}}$.

Arc-analyticity implies that
$F\bigl{(}t^{m_{1}},\dots,t^{m_{n}}\bigr{)}\in{\mathbb{R}}[[t]]$. We may
always assume that $(m_{1},\dots,m_{n})=1$, and then
${\mathbb{R}}[[t^{m_{1}},\dots,t^{m_{n}}]]\supset t^{r}{\mathbb{R}}[[t]]$ for
some $r=r(m_{1},\dots,m_{n})$. (See Definition 10 for explicit bounds on
$r(m_{1},\dots,m_{n})$.) So (1.1) states the vanishing of finitely many terms
of the Taylor expansion of $F\bigl{(}t^{m_{1}},\dots,t^{m_{n}}\bigr{)}$,
depending only on $m_{1},\dots,m_{n}$.

Proof. If $F$ is analytic then locally it can be written as a power series in
$x_{1},\dots,x_{n}$, so $F\bigl{(}t^{m_{1}},\dots,t^{m_{n}}\bigr{)}$ is a
power series in $t^{m_{1}},\dots,t^{m_{n}}$, as needed.

For the converse, assume that (1.1) holds. We define elementary monomial
coordinate changes in Definition 2, and note that (1.1) is preserved by them.
There are now 2 steps.

(Step 1.2) $F$ becomes analytic after a sequence of elementary monomial
coordinate changes. This is proved in Paragraph 8, using results of [Hir64]
and [BM90] on resolution of singularities.

Note that [Hir64] is used through the Equivariant Principalization Theorem 15,
but only for monomial ideals, which is a relatively easy special case.
However, the proof also uses Theorem 7, which in turn relies on Hironaka’s
uniformization theorem for subanalytic sets and his flattening theorem for
proper, real analytic maps [Hir73a, Hir73b]; these are considerably harder.

(Step 1.3) If $F$ satisfies (1.1) and becomes analytic after one elementary
monomial coordinate change, then $F$ is analytic. The 2 variable case is in
[KK22], the straightforward generalization is discussed in Paragraph 3.

We are now done by induction on the number of elementary monomial coordinate
changes. ∎

######  Definition 2 (Monomial coordinate changes).

By an elementary monomial coordinate change we mean: permuting the
coordinates, choosing $1<r\leq n$ and then setting
$\sigma:{\mathbb{R}}^{n}({\mathbf{x}^{\prime}})\to{\mathbb{R}}^{n}({\mathbf{x}})$
given by

$None$
$\sigma:(x^{\prime}_{1},\dots,x^{\prime}_{n})\mapsto(x^{\prime}_{1}x^{\prime}_{r},\dots,x^{\prime}_{r-1}x^{\prime}_{r},x^{\prime}_{r},\dots,x^{\prime}_{n}).$

Its inverse is defined for $(x_{r}\neq 0)$:

$None$
$x^{\prime}_{1}=x_{1}/x_{r},\dots,x^{\prime}_{r-1}=x_{r-1}/x_{r},x^{\prime}_{r}=x_{r},\dots,x^{\prime}_{n}=x_{n}.$

Comment. The $x^{\prime}_{i}$ are the usual coordinates on one of the charts
after blowing up $(x_{1}=\cdots=x_{r}=0)$. We are guided by blow-ups, but no
knowledge of them is needed if someone is willing to accept Theorem 5.

Thus $F^{\prime}:=F\circ\sigma$ is

$F^{\prime}(x^{\prime}_{1},\dots,x^{\prime}_{n}):=F(x^{\prime}_{1}x^{\prime}_{r},\dots,x^{\prime}_{r-1}x^{\prime}_{r},x^{\prime}_{r},\dots,x^{\prime}_{n}).$

If $\rho^{\prime}:t_{i}\mapsto(x^{\prime}_{i}:=t_{i}^{m_{i}})$ is a monomial
curve then its image

$\rho:t_{i}\mapsto\bigl{(}x_{1}=t^{m_{1}+m_{r}},\dots,x_{r-1}:=t^{m_{r-1}+m_{r}},x_{r}=t^{m_{r}},\dots,x_{n}=t^{m_{n}}\bigr{)}$

is a monomial curve and $F^{\prime}\circ\rho^{\prime}=F\circ\rho$. Thus if
(1.1) holds for $F\circ\rho$, then also holds for
$F^{\prime}\circ\rho^{\prime}$ (but not conversely).

Thus the validity of (1.1) for all monomial curves is preserved by elementary
monomial coordinate changes.

######  3Proof of Step 1.3.

Assume that there is an elementary coordinate change as in (2.1) such that

$F^{\prime}(x^{\prime}_{1},\dots,x^{\prime}_{n}):=F(x^{\prime}_{1}x^{\prime}_{r},\dots,x^{\prime}_{r-1}x^{\prime}_{r},x^{\prime}_{r},\dots,x^{\prime}_{n})$

is analytic. Then $F^{\prime}$ has a power series expansion involving
monomials

$(x^{\prime}_{1})^{a_{1}}\cdots(x^{\prime}_{n})^{a_{n}}=x_{1}^{a_{1}}\cdots
x_{r-1}^{a_{r-1}}x_{r}^{A_{r}}x_{r+1}^{a_{r+1}}\cdots x_{n}^{a_{n}},$

where $A_{r}(a_{1},\dots,a_{r})=a_{r}-(a_{1}+\cdots+a_{r-1})$. If
$A_{r}(a_{1},\dots,a_{r})\geq 0$ for every monomial in $F^{\prime}$, then
$F(x_{1},\dots,x_{n})$ is analytic as needed.

Otherwise there is a monomial
$M=(x^{\prime}_{1})^{a_{1}}\cdots(x^{\prime}_{n})^{a_{n}}$ in $F^{\prime}$ for
which $A_{r}<0$. As in Remark 4, choose
$(m^{\prime}_{1},\dots,m^{\prime}_{n})$ such that
$\textstyle{\sum}c_{i}m^{\prime}_{i}=\textstyle{\sum}a_{i}m^{\prime}_{i}$ has
no nontrivial solution in $c_{i}\in{\mathbb{N}}^{n}$. Set

$m_{1}:=m^{\prime}_{1}+m^{\prime}_{r},\dots,m_{r-1}:=m^{\prime}_{r-1}+m^{\prime}_{r},m_{r}=m^{\prime}_{r},\dots,m_{n}=m^{\prime}_{n}.$

We claim that (1.1) fails for this choice. Indeed, in
$F\bigl{(}t^{m_{1}},\dots,t^{m_{n}}\bigr{)}$ we get a monomial
$M\bigl{(}t^{m_{1}},\dots,t^{m_{n}}\bigr{)}$ of degree

$\textstyle{\sum}a_{i}m^{\prime}_{i}=-(a_{1}+\cdots+a_{r-1})m_{r}+\textstyle{\sum}a_{i}m_{i}.$

Any other monomial $(x^{\prime}_{1})^{c_{1}}\cdots(x^{\prime}_{n})^{c_{n}}$
gives different $t$-degree, so $M\bigl{(}t^{m_{1}},\dots,t^{m_{n}}\bigr{)}$ is
not canceled. Also,
$M\bigl{(}t^{m_{1}},\dots,t^{m_{n}}\bigr{)}\in{\mathbb{R}}[[t^{m_{1}},\dots,t^{m_{n}}]]$
iff

$\textstyle{\sum}c_{i}m_{i}=-(a_{1}+\cdots+a_{r-1})m_{r}+\textstyle{\sum}a_{i}m_{i}=\textstyle{\sum}a_{i}m^{\prime}_{i}$

has a solution in $c_{i}\in{\mathbb{N}}$. Equivalently,

$None$
$(c_{1}+\cdots+c_{r-1})m^{\prime}_{r}+\textstyle{\sum}c_{i}m^{\prime}_{i}=\textstyle{\sum}a_{i}m^{\prime}_{i}.$

On the left hand side the coefficient of $m^{\prime}_{r}$ is
$c_{1}+\cdots+c_{r-1}+c_{r}$, which is at least $c_{1}+\cdots+c_{r-1}$. On the
right hand side $a_{r}<a_{1}+\cdots+a_{r-1}$, since we assumed that
$A_{r}(a_{1},\dots,a_{r})=a_{r}-(a_{1}+\cdots+a_{r-1})<0$. Thus (3.1) would be
a nontrivial solution of
$\textstyle{\sum}c^{\prime}_{i}m^{\prime}_{i}=\textstyle{\sum}a_{i}m^{\prime}_{i}$,
a contradiction to our choice of the $(m^{\prime}_{1},\dots,m^{\prime}_{n})$.
Thus
$F\bigl{(}t^{m_{1}},\dots,t^{m_{n}}\bigr{)}\not\in{\mathbb{R}}[[t^{m_{1}},\dots,t^{m_{n}}]]$.∎

######  Remark 4.

Fix $a_{i}\in{\mathbb{N}}^{>0}$ and $0<s_{i}\in{\mathbb{R}}$ linearly
independent over ${\mathbb{Q}}$. Then
$\textstyle{\sum}c_{i}s_{i}=\textstyle{\sum}a_{i}s_{i}$ has no nontrivial
solution in $c_{i}\in{\mathbb{Q}}$. Moreover, there are only finitely many
$(b_{i})\in{\mathbb{N}}^{n}$ such that
$\bigl{|}\textstyle{\sum}b_{i}s_{i}-\textstyle{\sum}a_{i}s_{i}\bigr{|}<1$.
Thus there is an $\epsilon>0$ such that if $|s^{\prime}_{i}-s_{i}|<\epsilon$
then $\textstyle{\sum}c_{i}s^{\prime}_{i}=\textstyle{\sum}a_{i}s^{\prime}_{i}$
has no nontrivial solution in $c_{i}\in{\mathbb{N}}$. Choose $s^{\prime}_{i}$
rational and multiply through by the common denominator. We get
$m_{i}\in{\mathbb{N}}^{>0}$ such that
$\textstyle{\sum}c_{i}m_{i}=\textstyle{\sum}a_{i}m_{i}$ has no nontrivial
solution in $c_{i}\in{\mathbb{N}}$.

We have a lot of freedom in choosing the $m_{i}$, for example we can arrange
that all of them are primes.

The technical step of the proof is the following special case of [Kol19]. We
state it over an arbitrary field $K$, though for the current applications we
need only $K={\mathbb{R}}$. (Also, over finite fields we should work with
${\mathbb{A}}^{n}_{K}:=\operatorname{Spec}_{K}K[x_{1},\dots,x_{n}]$ instead on
$K^{n}$.)

######  Theorem 5 (Monomial principalization).

Let $K$ be a field and $h_{0},\dots,h_{r}\in K[[x_{1},\dots,x_{n}]]$ power
series. Then there is a sequence of elementary monomial coordinate changes
$\sigma_{i}:K^{n}\to K^{n}$ with composite
$\bar{\sigma}:K^{n}({\mathbf{z}})\to K^{n}({\mathbf{x}})$ given by
$\bar{\sigma}^{*}x_{i}=m_{i}({\mathbf{z}})$, and a monomial $M({\mathbf{z}})$,
such that the following hold.

  1. (5.1)

The
$\tilde{h}_{i}({\mathbf{z}}):=M({\mathbf{z}})^{-1}h_{i}\bigl{(}m_{1}({\mathbf{z}}),\dots,m_{n}({\mathbf{z}})\bigr{)}$
are power series, and

  2. (5.2)

$\tilde{h}_{j}(0,\dots,0)\neq 0$ for at least one $0\leq j\leq r$.

While this is a special case of [Kol19], in Paragraph 13 we give a proof that
uses only the statement of Hironaka’s Principalization Theorem, but very
little other algebraic geometry terminology or machinery.

######  Corollary 6 (Monomial regularization).

Let $h_{i}$ be real analytic functions on an open set ${\mathbf{0}}\in
U({\mathbf{x}})\subset{\mathbb{R}}^{n}({\mathbf{x}})$. Then there is a
sequence of elementary monomial coordinate changes
$\bar{\sigma}=\sigma_{1}\circ\cdots\circ\sigma_{s}:{\mathbb{R}}^{n}({\mathbf{z}})\to{\mathbb{R}}^{n}({\mathbf{x}})$
and an open set ${\mathbf{0}}\in
U({\mathbf{z}})\subset{\mathbb{R}}^{n}({\mathbf{z}})$, such that

$(h_{0}{:}\cdots{:}h_{r})\circ\bar{\sigma}:U({\mathbf{z}})\dasharrow{\mathbb{P}}^{r}$

is real analytic at the origin.

Note that this is different from Hironaka-type resolution of indeterminacies
in 2 ways. It is easier, since we want the composite to be real analytic at 1
point only, not everywhere. On the other hand, we are allowed to use only
elementary monomial coordinate changes, while the usual resolution methods
also use local analytic coordinate changes between successive blow-ups in an
essential way.

Proof. We apply Theorem 5 to the power series expansions of the $h_{i}$ to get
$\bar{\sigma}:{\mathbb{R}}^{n}({\mathbf{z}})\to{\mathbb{R}}^{n}({\mathbf{x}})$
and $\tilde{h}_{i}:=(h_{i}\circ\bar{\sigma})/M$ as in (5.1). The
$\tilde{h}_{i}$ are analytic at the origin, and $\tilde{h}_{j}$ is nonzero at
the origin for some $j$. Thus

$(h_{0}{:}\cdots{:}h_{r})\circ\bar{\sigma}:U({\mathbf{z}})\dasharrow{\mathbb{P}}^{r}$

agrees with

$\bigl{(}\tilde{h}_{0}{:}\cdots{:}\tilde{h}_{r}\bigr{)}:U({\mathbf{z}})\dasharrow{\mathbb{P}}^{r},$

and the latter is defined and analytic at the origin.∎

The following is proved, though differently stated, in [BM90, Thm.1.4].

######  Theorem 7.

Let $X$ be a real analytic manifold and $f:X\to{\mathbb{R}}$ an arc analytic
function whose graph is subanalytic. Then there is a sequence of real analytic
morphisms

$None$
$X_{m}\stackrel{{\scriptstyle\pi_{m}}}{{\longrightarrow}}\tilde{X}_{m-1}\stackrel{{\scriptstyle
u_{m-1}}}{{\longrightarrow}}X_{m-1}\stackrel{{\scriptstyle\pi_{m-1}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle\pi_{1}}}{{\longrightarrow}}\tilde{X}_{0}\stackrel{{\scriptstyle
u_{1}}}{{\longrightarrow}}X_{0}=X,$

where, for every $i$,

  1. (7.2)

$u_{i}:\tilde{X}_{i}\to X_{i}$ is a locally finite, open covering,

  2. (7.3)

$\pi_{i}:X_{i}\to\tilde{X}_{i-1}$ is the blow-up of a smooth, closed, real
analytic subset, and

  3. (7.4)

the composite $F_{m}:=F\circ u_{1}\circ\pi_{1}\circ\cdots\circ
u_{m-1}\circ\pi_{m}:X_{m}\to{\mathbb{R}}$ is real analytic. ∎

######  8Proof of Step 1.2.

Let $F:U_{0}\to{\mathbb{R}}$ be as in Theorem 1, and consider the sequence of
morphisms (7.1).

We prove that there is a sequence of elementary monomial coordinate changes
$\sigma_{i}:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ and an open neighborhood
$0\in U_{m}\subset{\mathbb{R}}^{n}$, such that

$\tau_{0}:=\sigma_{1}\circ\cdots\circ\sigma_{s(m)}:U_{m}\to{\mathbb{R}}^{n}=:X_{0}$

lifts to $\tau_{m}:U_{m}\to X_{m}$ that is real analytic at the origin. If
this holds then $F\circ\tau_{0}=F_{m}\circ\tau_{m}$ is real analytic at the
origin, as needed.

We construct the lifting $\tau_{m}$ by going up the chain (7.1). Thus assume
that we already have $\tau_{i}:U_{i}\to X_{i}$. Since $u_{i}:\tilde{X}_{i}\to
X_{i}$ is a locally finite, open covering, any $\tau_{i}:U_{i}\to X_{i}$ lifts
to $\tilde{\tau}_{i}:\tilde{U}_{i}\to\tilde{X}_{i}$ over some
$0\in\tilde{U}_{i}\subset U_{i}$.

It remains to lift each $\tilde{\tau}_{i}:\tilde{U}_{i}\to\tilde{X}_{i}$ to
$\tau_{i+1}:U_{i+1}\to X_{i+1}$.

Note that, in a neighborhood of $\tilde{\tau}_{i}({\mathbf{0}})$, the blow-up
$\pi_{i+1}$ is the closed graph of the meromorphic map

$(h_{0}^{(i)}{:}\cdots{:}h_{r_{i}}^{(i)}):\tilde{X}_{i}\dasharrow{\mathbb{P}}^{r_{i}},$

where the $h_{j}^{(i)}$ are real analytic. Consider the composite

$None$
$H^{(i)}:=\bigl{(}h_{0}^{(i)}\circ\tilde{\tau}_{i}{:}\cdots{:}h_{r_{i}}^{(i)}\circ\tilde{\tau}_{i}\bigr{)}:\tilde{U}_{i}\dasharrow{\mathbb{P}}^{r_{i}}.$

By Corollary 6 there is a sequence of elementary monomial coordinate changes
$\sigma^{(i)}:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ and
$U_{i+1}\subset{\mathbb{R}}^{n}$, such that
$H^{(i)}\circ\sigma^{(i)}:U_{i+1}\to{\mathbb{P}}^{r_{i}}$ is real analytic at
the origin. Then

$\tau_{i+1}:=\bigl{(}\tilde{\tau}_{i}\circ\sigma^{(i)},H^{(i)}\circ\sigma^{(i)}\bigr{)}:U_{i+1}\to
X_{i+1}\subset\tilde{X}_{i}\times{\mathbb{P}}^{r_{i}}$

is real analytic at the origin. ∎

## Further remarks and examples

######  Remark 9.

Even over ${\mathbb{C}}$, one can not check analyticity at a point using
bounded degree curves only.

To see this, let $h(x,y)$ be any irreducible polynomial of degree $n$. Set
$F=H(x,y)/h(x,y)$ where $H$ is any analytic function such that
$\operatorname{mult}_{(0,0)}H>n(n-1)$. As we discuss below, the restriction of
$F$ to every curve of degree $<n$ through the origin is analytic at the
origin, but $F$ is usually not.

To see this let $C:=(g=0)\subset{\mathbb{C}}^{2}$ be a curve of degree $d<n$.
Then $h$ does not vanish on $C$. Set $R:={\mathbb{C}}[x,y]/(g)$.
Multiplication by $h$ gives an isomorphism

$R[h^{-1}]/R\cong R/(h)\cong{\mathbb{C}}[x,y]/(g,h),$

and the latter has dimension $\leq nd$ by Bézout’s theorem. Both $x,y$ are
nilpotent on the summand supported at the origin, so $x^{i}y^{j}/h$ is regular
on $C$ at the origin whenever $i+j>nd$.

It would also not be enough to use bounded multiplicity (but unbounded degree)
monomial curves in Theorem 1. For example, set
$F_{m}=x^{m}y^{m}/(x^{2}+y^{2})$. Using Definition 10 we see that

$F_{m}(t^{m_{1}},t^{m_{2}})\in{\mathbb{C}}[[t^{m_{1}},t^{m_{2}}]]\quad\mbox{if}\quad\min\\{m_{1},m_{2}\\}\leq
m/2.$

On the other hand, it is clear from the proof of Theorem 1, especially
Paragraphs 3–4, that many sparse subsets of all monomial curves do work. For
example, as noted in Remark 4, it is enough to check (1.1) when the $m_{i}$
are all primes.

######  Definition 10 (Frobenius number).

The largest $m$ such that $t^{m}$ is not in $K[t^{m_{1}},\dots,t^{m_{n}}]$ is
called the Frobenius number of $\\{m_{1},\dots,m_{n}\\}$.

For $n=2$ the Frobenius number is $m_{1}m_{2}-m_{1}-m_{2}$. For $n\geq 3$ even
good estimates are hard, but it is known that the Frobenius number is at most
$(2/n)m_{1}m_{n}$ (assuming $m_{1}\leq\cdots\leq m_{n}$), see [EG72].

Thus if $F=g/h$ for some explicit power series $g,h$, then the condition (1.1)
is algorithmically decidable.

A purely algebraic version of Theorem 1 is the following.

######  Theorem 11.

Let $K$ be an arbitrary field and $f,g_{i}\in K[[x_{1},\dots,x_{n}]]$ power
series. Then $f\in(g_{1},\dots,g_{r})$ iff

$None$
$f(t^{m_{1}},\dots,t^{m_{n}})\in\bigl{(}g_{1}(t^{m_{1}},\dots,t^{m_{n}}),\dots,g_{r}(t^{m_{1}},\dots,t^{m_{n}})\bigr{)}$

in the ring $K[[t^{m_{1}},\dots,t^{m_{n}}]]$ for every
$0<m_{i}\in{\mathbb{N}}$.

Note that we need (11.1) in the ring $K[[t^{m_{1}},\dots,t^{m_{n}}]]$, not in
$K[[t]]$. This is a somewhat subtle condition, since
$K[[t^{m_{1}},\dots,t^{m_{n}}]]$ is not a principal ideal domain.

Requiring (11.1) in $K[[t]]$ would lead to the notion of integral dependence,
see [Laz04, SH06, LJT08].

This works over any field since principalization of monomial ideals is a
combinatorial question, hence independent of the characteristic, see [BM06].
However, Theorem 11 has a simpler, more direct proof using Gröbner bases.

## Proof of Theorem 5

We need the following special case of resolution of singularities. We discuss
its proof at the end.

######  Theorem 12 (Monomial principalization of monomial ideals).

Let $K$ be a field and $J\subset K[x_{1},\dots,x_{n}]$ a monomial ideal. Then
there is a sequence of elementary monomial coordinate changes
$\sigma_{i}:K^{n}\to K^{n}$ with composite
$\bar{\sigma}:K^{n}({\mathbf{z}})\to K^{n}({\mathbf{x}})$ given by
$\bar{\sigma}^{*}x_{i}=m_{i}({\mathbf{z}})$, such that
$\bar{\sigma}^{*}J\subset K[z_{1},\dots,z_{n}]$ is a principal ideal,
generated by a monomial.

######  13Proof of Theorem 5.

Let $E(h_{0},\dots,h_{r})\subset{\mathbb{N}}^{n}$ be the set of exponents
$(a_{1},\dots,a_{n})$ for which $x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$ appears in
some $h_{i}$ with nonzero coefficient.

The observation of [Kol19] is that one should work with the monomial ideal

$J=J(h_{0},\dots,h_{r}):=\bigl{(}x_{1}^{a_{1}}\cdots
x_{n}^{a_{n}}:(a_{1},\dots,a_{n})\in E(h_{0},\dots,h_{r})\bigr{)}\subset
K[x_{1},\dots,x_{n}].$

Next we apply Theorem 12 to get a composite of a sequence of elementary
monomial coordinate changes $\bar{\sigma}:K^{n}\to K^{n}$ such that
$\bar{\sigma}^{*}J$ is a principal ideal generated by

$M({\mathbf{z}})=m_{1}^{a_{1}}\cdots m_{n}^{a_{n}}\quad\mbox{for
some}\quad(a_{1},\dots,a_{n})\in E(h_{0},\dots,h_{r}).$

Also, if $x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$ appears in some $h_{j}$ with
nonzero coefficient, then $M({\mathbf{z}})$ appears in the same
$\bar{\sigma}^{*}h_{j}$ with nonzero coefficient.

Since $M({\mathbf{z}})$ divides each monomial in $\bar{\sigma}^{*}h_{i}$, the

$\tilde{h}_{i}({\mathbf{z}}):=M({\mathbf{z}})^{-1}h_{i}\bigl{(}m_{1}({\mathbf{z}}),\dots,m_{n}({\mathbf{z}})\bigr{)}$

are power series, and $\tilde{h}_{j}(0,\dots,0)\neq 0$. ∎

Next we discuss how to get Theorem 12 from the large literature on resolution
of singularities. First we outline how the method of Hironaka gives this,
though here we need to ‘follow the proof.’ Then we show how to derive Theorem
12 from the statement of the Equivariant Principalization Theorem.

######  14Hironaka’s method.

As in [Hir64], we start with an ideal $J:=(f_{1},\dots,f_{r})\subset
K[x_{1},\dots,x_{n}]$. We aim to produce a smooth subvariety $Z\subset K^{n}$
whose blow-up ‘simplifies’ the singularities of $J$. The procedure is rather
complicated, going through a sequence of new ideals with new generators.
However, all new generators are constructed by repeatedly applying 2 steps:

  * •

taking a partial derivative of a generator, or

  * •

multiplying 2 generators.

Both of these steps preserve monomials, so if we start with monomials $f_{i}$,
then all ideals in the Hironaka process are generated by monomials.

At the end we arrive at an ideal $J^{*}:=(f^{*}_{1},\dots,f^{*}_{s})\subset
K[x_{1},\dots,x_{n}]$ such that $J^{*}$ is the ideal of a smooth subvariety
$Z\subset K^{n}$.

The zero set of any monomial ideal is a union of intersections of coordinate
hyperplanes, and the only smooth ones are intersections of coordinate
hyperplanes. Thus $J^{*}$ defines an intersection of coordinate hyperplanes.
The corresponding blow-ups are exactly the elementary monomial coordinate
changes, as in Definition 2.

It turns out that we are in a much simpler situation than the one considered
by Hironaka, where it is quite difficult to establish that the process is
independent of the choice of the generators. In our case we have natural
generators.

Also, as shown in [BM06], the original method of Hironaka applies to monomial
ideals in any characteristic. The key change is that, instead of
differentiation $(cx^{m})^{\prime}=mcx^{m-1}$, one should use the operator
$cx^{m}\mapsto cx^{m-1}$. Then we get a process that works in positive
characteristic as well. ∎

Monomial ideals are usually dealt with as special cases of toric geometry. For
us the simplest explicit reference appears to be the following.

######  Theorem 15 (Equivariant Principalization).

Let $K$ be a field of characteristic 0. Let $G\curvearrowright X$ be a group
acting on a $K$-variety and $I\subset{\mathcal{O}}_{X}$ a $G$-invariant ideal
sheaf. Then there is a sequence

$(G\curvearrowright
X_{s})\stackrel{{\scriptstyle\pi_{s-1}}}{{\longrightarrow}}(G\curvearrowright
X_{s-1})\to\cdots\to(G\curvearrowright
X_{1})\stackrel{{\scriptstyle\pi_{0}}}{{\longrightarrow}}(G\curvearrowright
X_{0}=X)$

of blow-ups of $G$-invariant, smooth subvarieties, such that

$I_{s}:=(\pi_{0}\circ\cdots\circ\pi_{s})^{*}I\subset{\mathcal{O}}_{X_{s}}$

is locally principal. ∎

References. This is a special case of [Kol07, 3.26]. There is no group action
in the statement, but [Kol07, 3.26.2] says that the principalization given
there is ‘functorial on smooth morphisms.’ Isomorphisms are smooth morphisms,
so the principalization in [Kol07, 3.26] is invariant under all isomorphisms.

The theorem is slightly more general than [Hir64, p.142]; the improvements in
[BM97, EV98] imply the general case.

We aim to apply Theorem 15 to $X={\mathbb{A}}^{n}_{K}$ and $I\subset
K[x_{1},\dots,x_{n}]$ an ideal generated by monomials. We can choose the group
to be $(K^{\times})^{n}$.

######  Definition 16.

A standard toric chart over $K$ is $(T^{n}\curvearrowright K^{n})$, where we
view $K^{n}$ and $T^{n}:=(K^{\times})^{n}$ as $K$-varieties, and the action is
given by

$(\lambda_{1},\dots,\lambda_{n})\times(x_{1},\dots,x_{n})\mapsto(\lambda_{1}x_{1},\dots,\lambda_{n}x_{n}).$

We allow permuting the $x_{i}$ and changing the action by any
$(m_{ij})\in\mathrm{GL}_{n}({\mathbb{Z}})$ to

$(\lambda_{1},\dots,\lambda_{n})\times(x_{1},\dots,x_{n})\mapsto\bigl{(}\textstyle{\prod}_{j}\lambda_{j}^{m_{1j}}x_{1},\dots,\textstyle{\prod}_{j}\lambda_{j}^{m_{nj}}x_{n}\bigr{)}.$

A toric $K$-variety is a $(T^{n}\curvearrowright X)$, where $X$ is a smooth
$K$-variety of dimension $n$, that is covered by $T^{n}$-invariant, standard
charts.

Comment on terminology. If $K$ is algebraically closed, this is the usual
notion of smooth toric variety. If $K$ is not algebraically closed, many
authors allow replacing $T^{n}$ with some other $K$-form of it. Thus, over
${\mathbb{R}}$, one could use $({\mathbb{S}}^{1})^{n}$ instead. For us it is
better to work with $T^{n}=(K^{\times})^{n}$.

The prime example, and the only one we need, is the following.

######  Example 17.

Let $Z\subset K^{n}$ be the intersection of some coordinate hyperplanes, say
$(x_{1}=\cdots=x_{r}=0)$. The blow-up $B_{Z}K^{n}$ is toric. It is covered by
$r$ standard toric charts, each one an elementary monomial coordinate change.

The $r$th chart is described in Definition 2. We get the $i$th chart by
permuting the $1,\dots,r$ such that $i$ ends up in the last position, and then
using (2.1–2).

Note that the only $T^{n}$-invariant subvarieties on $(T^{n}\curvearrowright
K^{n})$ are the intersections of coordinate hyperplanes. Therefore, if
$(T^{n}\curvearrowright X)$ is a toric $K$-variety, and if $Z\subset X$ is a
$T^{n}$-invariant subvariety, then $(T^{n}\curvearrowright B_{Z}X)$ is also a
toric $K$-variety. Iterating these gives the following.

######  Lemma 18.

Let

$(T^{n}\curvearrowright
X_{s})\stackrel{{\scriptstyle\pi_{s-1}}}{{\longrightarrow}}(T^{n}\curvearrowright
X_{s-1})\to\cdots\to(T^{n}\curvearrowright
X_{1})\stackrel{{\scriptstyle\pi_{0}}}{{\longrightarrow}}(T^{n}\curvearrowright
K^{n})$

be a sequence of blow-ups of $T^{n}$-invariant subvarieties. Then $X_{s}$ is
covered by standard toric charts $X_{s}^{j}$, such that each $K^{n}\cong
X_{s}^{j}\to K^{n}$ is a composite of $\leq s$ elementary monomial coordinate
changes. ∎

######  19Proof of Theorem 12.

Combining Theorem 15 and Lemma 18 gives a sequence of elementary monomial
coordinate changes $\sigma_{i}:K^{n}\to K^{n}$ with composite
$\bar{\sigma}:K^{n}({\mathbf{z}})\to K^{n}({\mathbf{x}})$ given by
$\bar{\sigma}^{*}x_{i}=m_{i}({\mathbf{z}})$, such that
$\bar{\sigma}^{*}J\subset K[z_{1},\dots,z_{n}]$ is a principal ideal.

The pull-back of a monomial by a sequence of elementary monomial coordinate
changes is a monomial, so $\bar{\sigma}^{*}J\subset K[z_{1},\dots,z_{n}]$ is a
monomial ideal. Since every divisor of a monomial is a monomial, if a monomial
ideal $\bar{\sigma}^{*}J=(M_{1},\dots,M_{s})$ is principal, then its is
generated by one of the $M_{i}$. ∎

######  Comments 20 (Other principalization methods).

Principalization of monomial ideals is a combinatorial question, which can be
done in many different ways; see [KKMSD73, AMRT75, Cox00, GPT02, Tev07].

In our proof we rely on Step 1.3, thus we need a process that uses only
elementary blow-up sequences. The above papers are not designed to do this.

However, it would be possible to establish a version of Step 1.3 for an
arbitrary monomial coordinate change. Once that is done, one could use any
toric principalization method instead of Theorem 15.

######  Acknowledgments.

I thank W. Kucharz and K. Kurdyka for introducing me to this subject, and many
helpful comments and corrections. Partial financial support was provided by
the NSF under grant number DMS-1901855.

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