# Investigating Deep Neural Structures and their Interpretability in the
Domain of Voice Conversion

###### Abstract

Generative Adversarial Networks (GANs) are machine learning networks based
around creating synthetic data. Voice Conversion (VC) is a subset of voice
translation that involves translating the paralinguistic features of a source
speaker to a target speaker while preserving the linguistic information. The
aim of non-parallel conditional GANs for VC is to translate an acoustic speech
feature sequence from one domain to another without the use of paired data. In
the study reported here, we investigated the interpretability of state-of-the-
art implementations of non-parallel GANs in the domain of VC. We show that the
learned representations in the repeating layers of a particular GAN
architecture remain close to their original random initialised parameters,
demonstrating that it is the number of repeating layers that is more
responsible for the quality of the output. We also analysed the learned
representations of a model trained on one particular dataset when used during
transfer learning on another dataset. This showed extremely high levels of
similarity across the entire network. Together, these results provide new
insight into how the learned representations of deep generative networks
change during learning and the importance in the number of layers.111Audio
samples available at: https://samuelbroughton.github.io/interpretability-
demo-2020.

Index Terms—  Voice conversion (VC), generative adversarial networks (GANs),
canonical correlation analysis (CCA), SVCCA, non-parallel VC, multi-domain VC

##  1 Introduction

Deep Learning networks have been shown to exhibit superior abilities in a
range of problem domains [1, 2, 3]. However, such networks are _black-box
representations_ in terms of their interpretability [4], and this can mitigate
against informed decision making when selecting appropriate network
configurations.

One problem domain, voice conversion (VC), or voice style transfer, is a
technique aimed at modifying the linguistic style of speech while preserving
the linguistic information contained therein [5, 6, 7]. VC can be formulated
as a regression problem with the aim of building a function in which the
features of a source speaker A can be mapped to a target speaker B [8, 6, 7].
Applications of VC include modifying speaker identity in text-to-speech (TTS)
systems [9], aiding those with vocal disabilities [10] and generating accents
for assisted language conversion in domains such as real-time language
translation and device-assisted language learning [11].

Historically, methods employed to achieve VC have included mapping code books
[12], Gaussian mixture models (GMMs) [8, 9, 13] and artificial neural networks
(ANNs) [14, 15]. However, variations of generative adversarial networks (GANs)
[16] have recently shown success in a range of different domains, such as
producing convincingly real images and videos [2, 17], enhancing the quality
of images [1], generating new music [18] and, of interest here, a methodology
for achieving VC [19, 20, 21, 3, 22, 23].

Some of the VC methods mentioned above can be categorized as either parallel
or non-parallel. Parallel VC refers to source and target speaker utterances
being perfectly aligned [6, 7]. Such data can be a laborious task to collect.
Furthermore, once collected, the data would need to be pre-processed with
automatic time alignment which can fail, resulting in other methods of
correction. However, GANs are able to learn mapping functions between data of
similar domains and so mitigate the need for a parallel dataset [2]. Recent
state-of-the-art non-parallel generative VC architectures include CycleGAN-VC2
[3] and StarGAN-VC2 [21]. Both make use of a gated convolutional neural
network (CNN) [24], identity-mapping loss [25] and 2-1-2D CNN architecture
[3].

A major advantage of using the StarGAN [26] framework when compared to
CycleGAN [2], is the ability to perform multi-domain conversion whilst only
requiring a single generator. With regards to VC, the StarGAN framework allows
for learned mapping functions between multiple speakers. Extending this
framework, StarGAN-VC [27] and StarGAN-VC2 [21] make various modifications
including updates to the training objective and alterations to the network
architecture.

However, despite StarGAN-VC2 demonstrating superior VC in both objective and
subjective experiments when compared to StarGAN-VC [21], there has been very
little investigation into the interpretability of it’s network representations
- as is the case with many deep multi-layer neural networks, especially GANs
[28]. Being able understand how interpret deep multi-layer GANs would benefit
the development of new generative techniques and the improve the efficiency of
current methods, as interpretability studies have began to do so with
discriminative models [29, 30]. The motivation for this paper is to provide
some insight into the underlying generation process by focusing on the learned
network representations and network depth.

In this work, we conducted an evaluation of learned network representations by
performing Singular Vector Canonical Correlation Analysis (SVCCA) [30] in a
range of different experiments using an adaptation of the StarGAN-VC2 network.
The aim was to provide insight into the interpretability of GANs for VC by
addressing the similarity of optimally trained networks and their random
initial states. This was achieved by conducting experiments with networks
including frozen layers, observing how quickly networks reached their optimal
representations, exploring the effects of modifying the size of networks and
investigating learned network representations when trained using transfer
learning.

The rest of the paper is structured as follows: Section 2 outlines the
generative network architecture used, Section 3 discusses SVCCA and the
motivation to use it in this work, Section 4 describes the research questions
and experimental conditions, Section 5 discusses the results and probable
implications, and Section 6 draws the conclusion.

##  2 Generative Network Architecture

The network architecture implemented for the experiments presented in this
paper was based on StarGAN-VC2 [21], which allows for non-parallel many-to-
many learned mappings for VC.

###  2.1 Training objectives

The main objective of the StarGAN framework [26] is to learn many-to-many
mapping functions between multiple domains whilst only using a single
generator $G$. StarGAN does this by conditioning itself on ‘one-hot’
representations of domain codes $c\in\\{1,...,N\\}$, where $c$ and $N$
indicate the domain code the number of domains, respectively. More
specifically in StarGAN-VC2, $G$ can be formulated as the mapping function
$G(x,\hat{c})\xrightarrow{}\hat{x}$, taking an acoustic input feature sequence
$x\in\mathbb{R}^{Q\times T}$ and target domain code $\hat{c}$ to generate an
acoustic output feature sequence $\hat{x}$. StarGAN-VC2 does this by making
use of an adversarial loss [16], reconstruction or cycle-consistency loss [2]
and identity-mapping loss [25].

Adversarial loss is used in GANs to encourage generated data, conditioned on
target domain code, to be indistinguishable to that of real data [26]:

$\displaystyle\mathcal{L}_{adv}$ $\displaystyle=\mathbb{E}_{(x,c)\sim
P(x,c)}[\log D(x,c)]$ $\displaystyle+\mathbb{E}_{x\sim P(x),\hat{c}\sim
P(\hat{c})}[\log(1-D(G(x,\hat{c}),\hat{c}))],$ (1)

where $D$ is a real/fake discriminator that attempts to maximise this loss to
learn the decision boundary between real and fake features. $G$ attempts to
minimize this loss by generating an output indistinguishable to the real
acoustic features of target domain $\hat{c}$.

As discussed in the StarGAN-VC2 study [21], when handling both hard negative
and easy negative samples (e.g. same speaker domain conversion and different
speaker domain conversion), this condition can make it difficult to bring
generated output data close to real target data. Therefore, source-and-target
conditional adversarial loss [21] is used to help $G$ generate an output
closer to the real target data. However, during pre-experiments we found that
only using target domain input in $G$ and using both source-and-target domain
inputs in $D$ yielded a better output quality without affecting speaker
similarity. The modified source-and-target adversarial objective is defined
as:

$\displaystyle\mathcal{L}_{st\mathchar 45\relax adv}$
$\displaystyle=\mathbb{E}_{(x,c)\sim P(x,c),\hat{c}\sim P(\hat{c})}[\log
D(x,\hat{c},c)]$ $\displaystyle+\mathbb{E}_{(x,c)\sim P(x,c),\hat{c}\sim
P(\hat{c})}[\log D(G(x,\hat{c}),c,\hat{c})],$ (2)

Cycle-consistency loss is used in order to guarantee that the converted output
feature sequence preserves the source characteristics of input feature
sequence $x$ [2, 26]:

$\displaystyle\mathcal{L}_{cyc}=\mathbb{E}_{(x,c)\sim P(x,c),\hat{c}\sim
P(\hat{c})}[||x-G(G(x,\hat{c}),c)||_{1}].$ (3)

This cyclic constraint encourages $G$ to reconstruct the original input
feature $x$ from the generated output $\hat{x}$ and source domain code $c$.
This helps $G$ to preserve the linguistic information of the speech [27].

Identity-mapping loss is employed to encourage the preservation of input
feature identity within generated output data [25]:

$\displaystyle\mathcal{L}_{id}$ $\displaystyle=\mathbb{E}_{(x,c)\sim
P(x,c)}[||G(x,c)-x||].$ (4)

Identity-mapping loss has previously been used in image-to-image translation
for colour preservation [2].

The full objective can be summarised as follows:

$\displaystyle\mathcal{L}_{G}$ $\displaystyle=\mathcal{L}_{st\mathchar
45\relax adv}+\lambda_{cyc}\mathcal{L}_{cyc}+\lambda_{id}\mathcal{L}_{id},$
(5) $\displaystyle\mathcal{L}_{D}$ $\displaystyle=-\mathcal{L}_{st\mathchar
45\relax adv},$ (6)

where $\lambda_{cyc}$ and $\lambda_{id}$ are hyperparameters for each term.
Here, $G$ aims to minimise the loss whilst $D$ is trying to maximise it.

###  2.2 Network architectures

The fully convolutional GAN architecture used in the study reported here
allows for acoustic input feature sequences of arbitrary sizes.

Generator: The input to $G$ was an image of size $Q\times T$ of an acoustic
feature sequence $x$, where $Q$ and $T$ are the feature dimension and sequence
length, respectively. A 2-1-2D CNN [3, 21] architecture was used to construct
$G$. 2D convolutions are well suited for holding the original data structure
whilst 1D convolutions work well at dynamically changing the data [3]. The
implementation specifically used a gated CNN [24], which allowed for relevant
features to be selected and propagated based on previous layer states. The
effectiveness of a gated CNN for VC has already been confirmed in previous
studies [27, 20].

Conditional domain specific style code was injected in the 1D CNN architecture
by a modulation-based method [21]. Conditional instance normalisation (CIN)
[31, 32] was used to modulate parameters in a domain-specific manner:

$\displaystyle\text{CIN}(f,\hat{c})=\gamma_{\hat{c}}(\frac{f-\mu(f)}{\sigma(f)})+\beta_{\hat{c}},$
(7)

where $\mu(f)$ and $\sigma(f)$ are the average and standard deviation of
feature $f$ and $\gamma_{\hat{c}}$ and $\beta_{\hat{c}}$ are domain-specific
scale and bias parameters, respectively.

The 1D repeating blocks were not residual because the use of skip connections
was reported to result in partial conversion [33].

Real/Fake Discriminator: A 2D gated CNN [24] was used for the architecture of
the real/fake discriminator $D$, which has been formulated as a projection
discriminator [34], as seen in StarGAN-VC2 [21]. $D$ outputs a sequence of
probabilities, calculating how close the input acoustic feature sequence $x$
is to domain $c$.

Fig. 1: Network architectures of the fully convolutional [35] generator and
discriminator based on StarGAN-VC2 [21]. In the input, output and reshape
layers ‘h’, ‘w’ and ‘c’ represent the height, width and channel number
respectively. In the Conv2d, Conv1D and ConvT2D convolution layers, ‘k’, ‘c’
and ‘s’ represent the kernel size, channel number and stride, respectively.
‘IN’, ‘GLU’, ‘GSP’ and ‘FC’ denote instance normalisation [36], gated linear
unit [24], global sum pooling and fully connected layers, respectively. $N=9$
repeating 1D CNN blocks were used for all experiments unless otherwise
specified.

##  3 SVCCA on Deep Neural Representations

Singular Vector Canonical Correlation Analysis (SVCCA) is an extension of
Canonical Correlation Analysis (CCA), a method used in statistics to measure
the similarity of two vectors formed by some underlying process [37, 38, 39].
In the case of deep neural networks, these are the “neuron activation vectors”
formed from training on a particular dataset [39, 30]. A single neuron
activation vector is the output of a single neuron of a layer in a network.
Combining the outputs of all neurons for a particular layer in a network
results in a set of multidimensional output [39, 30]. Subsequently, CCA can be
used to compare the similarity between two layers of the same network, similar
networks using layers of same/differing dimensionality, or a given layer at
different stages of training [39].

SVCCA is an extension to CCA that involves a preprocessing step [39, 30]. The
authors of [30] explain that SVCCA takes the same inputs as CCA, for example
two layers of a neural network $l_{1}$ and $l_{2}$ that each contain a set of
neuron activation vectors. SVCCA then factorises the vectors by computing
Singular Value Decomposition (SVD) over each layer to obtain subspaces
$l^{\prime}_{1}\subset l_{1}$ and $l^{\prime}_{2}\subset l_{2}$. These
subspaces contain the most important variance directions, which can account
for 99% of the variance in input layers $l_{1}$ and $l_{2}$ [30]. CCA is then
performed on $l^{\prime}_{1}$ and $l^{\prime}_{2}$ to return the correlation
coefficients, providing a measure of similarity of the two layers.

The motivation for SVCCA in this paper is to provide a similarity metric for
the comparison of various layers in the generator network. This allows for the
interpretation of learned network representations at different stages of
training.

##  4 Experiments

Datasets: To evaluate our methods, we made use of the Device and Produced
Speech Dataset [40], as seen in the multi-speaker VC task in the Voice
Conversion Challenge 2018 (VCC2018) [7] and the English Multi-speaker Corpus
for CSTR Voice Cloning Toolkit (VCTK) [41]. We used a subset of both datasets
in all experiments except during transfer learning where the initial model was
trained using the VCTK dataset.

In both datasets four speakers were selected covering all inter- and intra-
gender conversions. In the VCTK dataset we selected speakers labelled p229,
p236, p232 and p243; speakers p229 and p236 are female, and speakers p232 and
p243 are male. The data from VCC2018 mimicked the data used to test StarGAN-
VC2 [21], whereby VCC2SF1 and VCC2SF2 are female speakers, and VCC2SM1 and
VCC2SM2 are male speakers. Speakerwise normalisation was conducted as a pre-
process.

For each experiment, $4\times 3=12$ source-and-target pair mappings were
learnt for each single model trained on both datasets. All the recordings for
both datasets were downsampled to 22.05 kHz. 36 Mel-cepstral coefficients
(MCEPs) were extracted from each recording. The logarithmic fundamental
frequency ($\log F_{0}$) and aperiodicities (APs) were extracted every 5 ms
using the WORLD vocoder [42].

Conversion process: The conversion process mimicked that of StarGAN-VC [27]
and StarGAN-VC2 [21], by not using any form of post filtering [43, 44] or
powerful vocoding [45, 46] and just focusing on MCEP conversion222Audio
samples available at: https://samuelbroughton.github.io/interpretability-
demo-2020.. As in previous studies, the WORLD vocoder [42] was used to
synthesise speech, directly taking APs and converting the $\log F_{0}$ using a
logarithm Gaussian normalised transformation [47].

Network implementations: Figure 1 presents the network architectures for $G$
and $D$, influenced by StarGAN-VC2 [21] and CycleGAN-VC2 [3]. The networks
were initially trained for $2\times 10^{5}$ batch iterations on both datasets.
During transfer learning, optimal models trained on the VCTK dataset were
selected and trained for an extra $1\times 10^{5}$ batch iterations on the
VCC2018 dataset. During the training of the networks for all experiments, the
states for $G$ and $D$ were saved at every $1\times 10^{4}$ batch iterations.
All networks were trained using the Adam optimizer [48] with a momentum term
$\beta_{1}$ set to $0.5$. The batch size was set to $16$ and we randomly
cropped segments of 512 frames from randomly selected sentences. Learning
rates for $G$ and $D$ were both set to $0.0001$, $\lambda_{cyc}=10$ and
$\lambda_{id}=5$. $\mathcal{L}_{id}$ was only used for the first $10^{4}$
iterations. $9$ repeating 1D CNN blocks were used for all experiments unless
otherwise specified. Least squares GAN [49] was used for a GAN objective.

Experimental investigation: Experiments were conducted in order to provide
insights into questions relating to the interpretability of the trained
networks. Experiment 1 addressed the issue as to how similar the learned
representations of the optimally trained network are to its random
initialisation. Experiment 2 addressed the question of how similar the learned
representations of networks trained via transfer learning on a new dataset are
to their previously optimal representations when trained on the original
dataset. Experiment 3 addressed the issue as to how similar the learned
representations of networks with various frozen repeating layers are.
Experiment 4 addressed the question of how the quality of the output feature
sequence changes with networks of a differing number of repeating layers.

##  5 Results and Discussions

0.10.30.50.70.9$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$Batch Iteration ($\times
10^{5}$)CCA DistanceD-sampling layersRepeat layersU-sampling layers Fig. 2:
Average CCA distance between the downsampling, repeating and upsampling
portions of the trained network and its random initial states (trained on the
VCTK dataset).
D1D2D3DCR1R2R3R4R5R6R7R8R9UCU1U2Out$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$LayerCCA
Distance$0.1\times 10^{5}$ BIs$0.5\times 10^{5}$ BIs$1\times 10^{5}$ BIs Fig.
3: CCA distance between each layer of a network at different stages of
training and its random initial states, where “BI” denotes batch iteration
trained on the VCTK dataset.
D1D2D3DCR1R2R3R4R5R6R7R8R9UCU1U2Out$0.8$$0.85$$0.9$$0.95$$1$LayerCCA
Distance$0.5\times 10^{3}$ BIs$1\times 10^{4}$ BIs Fig. 4: CCA distance
between each layer of a network at different stages of training during
transfer learning from the initial states of the previous optimally trained
network. Transfer learning was conducted using the VCC2018 dataset from a
network previously trained on the VCTK dataset. “BI” denotes the number of
batch iterations trained.
D1D2D3DCR1R2R3R4R5R6R7R8R9UCU1U2Out$0$$0.2$$0.4$$0.6$$0.8$$1$LayerCCA
DistanceNetwork ANetwork BNetwork C Fig. 5: CCA distance between each layer of
three different optimally trained networks with varying frozen layers and an
optimally trained network with no frozen layers. Network A was trained with
the parameters of layers R2 and R3 frozen, network B was trained with the
parameters of layers R4 and R5 frozen, and network C was trained with the
parameters of layers R6, R7 and R8 frozen. All three networks are extremely
similar in terms of their acoustic output when compared with the same
optimally trained network with no layers frozen.

Experiment 1: To assess how close the optimally trained network’s learned
representations were to their random initialisations, SVCCA was used to
compare networks at 0 and their optimal number of batch iterations. The number
of optimal batch iterations for networks trained on the VCTK and VCC2018
datasets were found to be approximately $1\times 10^{5}$ and $0.9\times
10^{5}$, respectively.

Figures 2 and 3 show the CCA distance between the learned representations of
layers in the network at different stages of training and their random
initialisation. Both figures show a greater correlation of similarity in the
learned network representations of the repeating 1D CNN layers (R1-R9) and
their random initial states when compared to the less similar downsampling and
upsampling portions of the network. Both figures show networks trained using
the VCTK dataset however, similar results were observed with the VCC2018
dataset.

The extreme similarity observed at D1 can be seen as a fundamental trait of
these networks. During pre-experiments, this trait was also seen in training
StarGAN-VC [27]. We removed the GLU of the first downsampling layer to check
if this was preventing the first convolution from learning as much as it
could. However, the same trait was still observed.

The results show that the learned network representations of these optimally
trained networks remain close to their initial random states, especially in
the repeating 1D CNN layers, which are reportedly responsible for the main
conversion process [3].

Experiment 2: The optimal model trained on the VCTK dataset in experiment 1
was used as the initial state for training during transfer learning on the
VCC2018 dataset. Figure 4 shows that the learned parameters of the entire
network remained extremely close to its initial network representations (the
parameters learned from training on the VCTK dataset). The model converged
after approximately $6000$ batch iterations and suffered from partial modal
collapse.

The similarity between target reference and converted synthesised samples
transfer learning model was poor when compared with the original model trained
on the same dataset. This could be due to the difference in speaker regions
across datasets.

Experiment 3: The random initial state of the models trained in experiment 1
were used to train networks with various frozen layers in the repeating
portion of the network. A total of three networks were evaluated with various
frozen layers, the first of which froze R2 and R3. The second network froze R4
and R5, and the third network froze R6, R7 and R8. Figure 5 shows the
similarity of these networks when compared against the optimally trained model
from experiment 1. The repeating 1D layers again showed a high degree of
similarity in their learned network representations. All networks were
extremely similar in terms of their acoustic output when compared with the
optimally trained model.

Experiment 4: The random initial state of the models trained in experiment 1
were used to train networks with differing numbers of repeating 1D layers. Six
models were trained with 3, 5, 7, 11, 13 and 15 repeating layers in addition
to the previously trained model from experiment 1, which had 9 repeating
layers. It was observed that, in general, the audio quality of the models
using 3, 5, 7 and 9 repeating layers sounded better than the models using 11,
13 and 15 layers. However, each model included at least one instance of having
a worse quality of output than their counterparts for various different
source-target pairs.

It was also observed that, as the number of repeating layers increased, the
modification of speaker identity was more pronounced. In other words, the
output from models with a greater number of repeating layers had clearer
accents than the output of those with fewer repeating layers. However, at some
points the modification of speaker identity was so pronounced that the
intelligibility of the audio began to deteriorate. Also, as the amount of
repeating layers of the network increased, so the overall level of noise
increased. Networks consisting of 3 and 5 repeating layers struggled to
converge whilst networks using 11, 13 and 15 suffered from a vanishing
gradient.

##  6 Conclusions

In the research reported here, we provide new insights into the
interpretability of Generative Adversarial Networks (GANs) for Voice
Conversion (VC). Using a network architecture based on StarGAN-VC2 [21], we
conducted an investigation into the learned representations of the network
over a range of different experimental conditions. The results showed that
there is at least one local optimum that lies close to the random initial
states of the network. It was also found that it is the number of repeating
layers in the network architecture that has a noticeable effect on the quality
of the output speech. In general, as the number of repeating layers in the
network increased, so too did the noise and certain aspects of speaker
identity became more pronounced. Future work will involve looking more into
the importance of network depth in GANs for VC.

## References

  * [1] Xintao Wang, Ke Yu, Shixiang Wu, Jinjin Gu, Yihao Liu, Chao Dong, Yu Qiao, and Chen Change Loy,  “Esrgan: Enhanced super-resolution generative adversarial networks,”  in Proceedings of the European Conference on Computer Vision (ECCV), 2018, pp. 0–0.
  * [2] Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros,  “Unpaired image-to-image translation using cycle-consistent adversarial networks,”  in Proceedings of the IEEE international conference on computer vision, 2017, pp. 2223–2232.
  * [3] Takuhiro Kaneko, Hirokazu Kameoka, Kou Tanaka, and Nobukatsu Hojo,  “Cyclegan-vc2: Improved cyclegan-based non-parallel voice conversion,”  in ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2019, pp. 6820–6824.
  * [4] Quan-shi Zhang and Song-Chun Zhu,  “Visual interpretability for deep learning: a survey,”  Frontiers of Information Technology & Electronic Engineering, vol. 19, no. 1, pp. 27–39, 2018.
  * [5] Seyed Hamidreza Mohammadi and Alexander Kain,  “An overview of voice conversion systems,”  Speech Communication, vol. 88, pp. 65–82, 2017.
  * [6] Tomoki Toda, Ling-Hui Chen, Daisuke Saito, Fernando Villavicencio, Mirjam Wester, Zhizheng Wu, and Junichi Yamagishi,  “The voice conversion challenge 2016,”  in Interspeech, 2016, pp. 1632–1636.
  * [7] Jaime Lorenzo-Trueba, Junichi Yamagishi, Tomoki Toda, Daisuke Saito, Fernando Villavicencio, Tomi Kinnunen, and Zhenhua Ling,  “The voice conversion challenge 2018: Promoting development of parallel and nonparallel methods,”  arXiv preprint arXiv:1804.04262, 2018.
  * [8] Yannis Stylianou, Olivier Cappé, and Eric Moulines,  “Continuous probabilistic transform for voice conversion,”  IEEE Transactions on speech and audio processing, vol. 6, no. 2, pp. 131–142, 1998.
  * [9] Hiromichi Kawanami, Yohei Iwami, Tomoki Toda, Hiroshi Saruwatari, and Kiyohiro Shikano,  “Gmm-based voice conversion applied to emotional speech synthesis,”  in Eighth European Conference on Speech Communication and Technology, 2003.
  * [10] Keigo Nakamura, Tomoki Toda, Hiroshi Saruwatari, and Kiyohiro Shikano,  “Speaking-aid systems using gmm-based voice conversion for electrolaryngeal speech,”  Speech Communication, vol. 54, no. 1, pp. 134–146, 2012.
  * [11] Daniel Felps, Heather Bortfeld, and Ricardo Gutierrez-Osuna,  “Foreign accent conversion in computer assisted pronunciation training,”  Speech communication, vol. 51, no. 10, pp. 920–932, 2009.
  * [12] Masanobu Abe, Satoshi Nakamura, Kiyohiro Shikano, and Hisao Kuwabara,  “Voice conversion through vector quantization,”  Journal of the Acoustical Society of Japan (E), vol. 11, no. 2, pp. 71–76, 1990.
  * [13] Tomoki Toda, Alan W Black, and Keiichi Tokuda,  “Voice conversion based on maximum-likelihood estimation of spectral parameter trajectory,”  IEEE Transactions on Audio, Speech, and Language Processing, vol. 15, no. 8, pp. 2222–2235, 2007.
  * [14] Srinivas Desai, E Veera Raghavendra, B Yegnanarayana, Alan W Black, and Kishore Prahallad,  “Voice conversion using artificial neural networks,”  in 2009 IEEE International Conference on Acoustics, Speech and Signal Processing. IEEE, 2009, pp. 3893–3896.
  * [15] Seyed Hamidreza Mohammadi and Alexander Kain,  “Voice conversion using deep neural networks with speaker-independent pre-training,”  in 2014 IEEE Spoken Language Technology Workshop (SLT). IEEE, 2014, pp. 19–23.
  * [16] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio,  “Generative adversarial nets,”  in Advances in neural information processing systems, 2014, pp. 2672–2680.
  * [17] Tero Karras, Samuli Laine, and Timo Aila,  “A style-based generator architecture for generative adversarial networks,”  in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2019, pp. 4401–4410.
  * [18] Hao-Wen Dong, Wen-Yi Hsiao, Li-Chia Yang, and Yi-Hsuan Yang,  “Musegan: Multi-track sequential generative adversarial networks for symbolic music generation and accompaniment,”  in Thirty-Second AAAI Conference on Artificial Intelligence, 2018\.
  * [19] Chin-Cheng Hsu, Hsin-Te Hwang, Yi-Chiao Wu, Yu Tsao, and Hsin-Min Wang,  “Voice conversion from unaligned corpora using variational autoencoding wasserstein generative adversarial networks,”  arXiv preprint arXiv:1704.00849, 2017.
  * [20] Takuhiro Kaneko, Hirokazu Kameoka, Kaoru Hiramatsu, and Kunio Kashino,  “Sequence-to-sequence voice conversion with similarity metric learned using generative adversarial networks.,”  in INTERSPEECH, 2017, vol. 2017, pp. 1283–1287.
  * [21] Takuhiro Kaneko, Hirokazu Kameoka, Kou Tanaka, and Nobukatsu Hojo,  “Stargan-vc2: Rethinking conditional methods for stargan-based voice conversion,”  arXiv preprint arXiv:1907.12279, 2019.
  * [22] Dipjyoti Paul, Yannis Pantazis, and Yannis Stylianou,  “Non-parallel voice conversion using weighted generative adversarial networks,”  Proc. Interspeech 2019, pp. 659–663, 2019.
  * [23] Hung-Yi Lee and Yu Tsao,  “Generative adversarial network and its applications to speech processing and natural language processing,” Speech Processing Laboratory, National Taiwan University. http://speech.ee.ntu.edu.tw/~tlkagk/GAN_3hour.pdf,  (Accessed: November 19, 2019).
  * [24] Yann N Dauphin, Angela Fan, Michael Auli, and David Grangier,  “Language modeling with gated convolutional networks,”  in Proceedings of the 34th International Conference on Machine Learning-Volume 70. JMLR. org, 2017, pp. 933–941.
  * [25] Yaniv Taigman, Adam Polyak, and Lior Wolf,  “Unsupervised cross-domain image generation,”  arXiv preprint arXiv:1611.02200, 2016.
  * [26] Yunjey Choi, Minje Choi, Munyoung Kim, Jung-Woo Ha, Sunghun Kim, and Jaegul Choo,  “Stargan: Unified generative adversarial networks for multi-domain image-to-image translation,”  in Proceedings of the IEEE conference on computer vision and pattern recognition, 2018, pp. 8789–8797.
  * [27] Hirokazu Kameoka, Takuhiro Kaneko, Kou Tanaka, and Nobukatsu Hojo,  “Stargan-vc: Non-parallel many-to-many voice conversion using star generative adversarial networks,”  in 2018 IEEE Spoken Language Technology Workshop (SLT). IEEE, 2018, pp. 266–273.
  * [28] Andrey Voynov and Artem Babenko,  “Rpgan: Gans interpretability via random routing,”  arXiv preprint arXiv:1912.10920, 2019.
  * [29] Aravindh Mahendran and Andrea Vedaldi,  “Understanding deep image representations by inverting them,”  in Proceedings of the IEEE conference on computer vision and pattern recognition, 2015, pp. 5188–5196.
  * [30] Maithra Raghu, Justin Gilmer, Jason Yosinski, and Jascha Sohl-Dickstein,  “Svcca: Singular vector canonical correlation analysis for deep learning dynamics and interpretability,”  in Advances in Neural Information Processing Systems, 2017, pp. 6076–6085.
  * [31] Vincent Dumoulin, Jonathon Shlens, and Manjunath Kudlur,  “A learned representation for artistic style,”  arXiv preprint arXiv:1610.07629, 2016.
  * [32] Xun Huang and Serge Belongie,  “Arbitrary style transfer in real-time with adaptive instance normalization,”  in Proceedings of the IEEE International Conference on Computer Vision, 2017, pp. 1501–1510.
  * [33] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun,  “Deep residual learning for image recognition,”  in Proceedings of the IEEE conference on computer vision and pattern recognition, 2016, pp. 770–778.
  * [34] Takeru Miyato and Masanori Koyama,  “cgans with projection discriminator,”  arXiv preprint arXiv:1802.05637, 2018.
  * [35] Jonathan Long, Evan Shelhamer, and Trevor Darrell,  “Fully convolutional networks for semantic segmentation,”  in Proceedings of the IEEE conference on computer vision and pattern recognition, 2015, pp. 3431–3440.
  * [36] Dmitry Ulyanov, Andrea Vedaldi, and Victor Lempitsky,  “Instance normalization: The missing ingredient for fast stylization,”  arXiv preprint arXiv:1607.08022, 2016.
  * [37] Harold Hotelling,  “Relations between two sets of variates,”  in Breakthroughs in statistics, pp. 162–190. Springer, 1992.
  * [38] David R Hardoon, Sandor Szedmak, and John Shawe-Taylor,  “Canonical correlation analysis: An overview with application to learning methods,”  Neural computation, vol. 16, no. 12, pp. 2639–2664, 2004.
  * [39] Ari Morcos, Maithra Raghu, and Samy Bengio,  “Insights on representational similarity in neural networks with canonical correlation,”  in Advances in Neural Information Processing Systems, 2018, pp. 5727–5736.
  * [40] Gautham J Mysore,  “Can we automatically transform speech recorded on common consumer devices in real-world environments into professional production quality speech?—a dataset, insights, and challenges,”  IEEE Signal Processing Letters, vol. 22, no. 8, pp. 1006–1010, 2014\.
  * [41] Christophe Veaux, Junichi Yamagishi, and Kirsten MacDonald,  “Cstr vctk corpus: English multi-speaker corpus for cstr voice cloning toolkit, [sound],” Datashare, Edinburgh. https://doi.org/10.7488/ds/1994,  (Accessed: December 16, 2019).
  * [42] Masanori Morise, Fumiya Yokomori, and Kenji Ozawa,  “World: a vocoder-based high-quality speech synthesis system for real-time applications,”  IEICE TRANSACTIONS on Information and Systems, vol. 99, no. 7, pp. 1877–1884, 2016.
  * [43] Takuhiro Kaneko, Hirokazu Kameoka, Nobukatsu Hojo, Yusuke Ijima, Kaoru Hiramatsu, and Kunio Kashino,  “Generative adversarial network-based postfilter for statistical parametric speech synthesis,”  in 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2017, pp. 4910–4914.
  * [44] Takuhiro Kaneko, Shinji Takaki, Hirokazu Kameoka, and Junichi Yamagishi,  “Generative adversarial network-based postfilter for stft spectrograms.,”  in INTERSPEECH, 2017, pp. 3389–3393.
  * [45] Aaron van den Oord, Sander Dieleman, Heiga Zen, Karen Simonyan, Oriol Vinyals, Alex Graves, Nal Kalchbrenner, Andrew Senior, and Koray Kavukcuoglu,  “Wavenet: A generative model for raw audio,”  arXiv preprint arXiv:1609.03499, 2016.
  * [46] Akira Tamamori, Tomoki Hayashi, Kazuhiro Kobayashi, Kazuya Takeda, and Tomoki Toda,  “Speaker-dependent wavenet vocoder.,”  in Interspeech, 2017, vol. 2017, pp. 1118–1122.
  * [47] Kun Liu, Jianping Zhang, and Yonghong Yan,  “High quality voice conversion through phoneme-based linear mapping functions with straight for mandarin,”  in Fourth International Conference on Fuzzy Systems and Knowledge Discovery (FSKD 2007). IEEE, 2007, vol. 4, pp. 410–414.
  * [48] Diederik P Kingma and Jimmy Ba,  “Adam: A method for stochastic optimization,”  arXiv preprint arXiv:1412.6980, 2014.
  * [49] Xudong Mao, Qing Li, Haoran Xie, Raymond YK Lau, Zhen Wang, and Stephen Paul Smolley,  “Least squares generative adversarial networks,”  in Proceedings of the IEEE international conference on computer vision, 2017, pp. 2794–2802.


# A limiting result for the Ramsey theory of functional equations

Paulo Henrique Arruda  Fakultät für Mathematik
Universität Wien
Oskar-Morgenstern-Platz 1
1090 Vienna, Austria<EMAIL_ADDRESS>and  Lorenzo Luperi Baglini
Dipartimento di Matematica
Università di Milano
Via Saldini 50
20133 Milano, Italy<EMAIL_ADDRESS>
###### Abstract.

We study systems of functional equations whose solutions can be parameterized
in function of one of the variables; our main result proves that the partition
regularity (PR) of such systems can be completely characterized by the
existence of constant solutions. As applications of this result, we prove the
following:

  * •

A complete characterization of PR systems of Diophantine equations in two
variables over $\mathbb{N}$. In particular, we prove that the only infinitely
PR irreducible equation in two variables is $x=y$;

  * •

PR of $S$-unit equations and the failure of Rado’s Theorem for finitely
generated multiplicative subgroups of $\mathbb{C}$; and

  * •

a complete characterization of the PR of two classes of polynomial exponential
equations.

###### Key words and phrases:

partition regularity of equations; ultrafilters; Ramsey theory

###### 2020 Mathematics Subject Classification:

Primary 05D10, 11B75; Secondary 11D72, 11D61, 54D80.

##  1\. Introduction

This paper deals with the study of the so-called _partition regularity of
systems of equations_.

######  Definition 1.1.

Let $S$ be an infinite set, let $n,m\in\mathbb{N}$ and let
$f_{1},\dots,f_{m}:S^{n}\to S$. Let

(1.1)
$\sigma(x_{1},\dots,x_{n})=\left\\{\begin{matrix}f_{1}(x_{1},\dots,x_{n}),\\\
\vdots\\\ f_{m}(x_{1},\dots,x_{n}).\end{matrix}\right.$

Given $\boldsymbol{s}=(s_{1},\dots,s_{m})\in S^{m}$, we say that the system
$\sigma\left(x_{1},\dots,x_{n}\right)=\boldsymbol{s}$ is partition regular
(abbreviated as PR from now on) on $A\subseteq S$ if for all $r\in\mathbb{N}$
and all $c:S\to\\{1,\dots,r\\}$, there are $i\in\\{1,\dots,r\\}$ and
$a_{1},\dots,a_{n}\in c_{i}^{-1}(A)$ such that
$\sigma(a_{1},\dots,a_{n})=\boldsymbol{s}$.

Whenever the context is clear, we write
$\sigma\left(x_{1},\dots,x_{n}\right)\allowbreak=\boldsymbol{s}$ to mean that
it is as in (1.1). When the system consists just of one equation, we will say
that the equation is PR; if the equation has the form
$P\left(x_{1},\dots,x_{n}\right)=0$ where $P$ is a polynomial, we say also
that $P$ is PR to mean that the equation $P\left(x_{1},\dots,x_{n}\right)=0$
is PR. Functions $c:A\to\\{1,\dots,r\\}$ are usually referred to as colorings,
and the condition $c(a_{1})=\dots=c(a_{n})$ is usually rephrased as
$a_{1},\dots,a_{n}$ are $c$-monochromatic. Given a $k\in\mathbb{N}$, $[k]$
denotes the set $\\{1,\dots,k\\}$.

It is well known that the study of PR properties is intertwined with the study
of the ultrafilters over a set $S$. The monograph [10] provides an
introduction to the topic and we assume that the reader has familiarity with
the basic theory of ultrafilters. This relationship is made explicit by the
following result, see [6, Proposition 1.8] or, more generally, [10, Theorem
3.11]:

######  Theorem 1.2.

Under the hypothesis of Definition 1.1, the system (1.1) is partition regular
over $S$ if and only if there is an ultrafilter $\mathcal{U}$ on $\beta S$
such that for all $A\in\mathcal{U}$ there are $a_{1},\dots,a_{n}\in A$
satisfying $\sigma\left(a_{1},\dots,a_{n}\right)=\boldsymbol{s}$.

An ultrafilter $\mathcal{U}$ with the property of Theorem 1.2 will be called a
_witness_ of the partition regularity of
$\sigma\left(x_{1},\dots,x_{n}\right)=0$.

The general problem of the partition regularity of systems of Diophantine
equations, especially111To avoid trivialities, in this paper we convene that
$\mathbb{N}=\\{1,2,3,\dots\\}$. on $\mathbb{N}$, has long been studied. The
linear case was settled by R. Rado in 1933 in terms of the so-called columns
condition:

######  Definition 1.3.

Let $R$ be a commutative ring, $A$ be a $m\times n$ matrix with entries in $R$
and $C_{1},\dots,C_{m}$ be the columns of $A$; we say that $A$ satisfies the
columns condition if there is a partition $I_{0},\dots,I_{r}$ of $[m]$ such
that

  1. (1)

$\sum_{i\in I_{0}}C_{i}=\boldsymbol{0}$; and

  2. (2)

given any $u\in[r]$, $\sum_{i\in
I_{u}}C_{i}\in\operatorname{span}_{K}\\{C_{j}\mid j\in I_{0}\cup\dots\cup
I_{u-1}\\}$, where $K$ is the field of fractions of $R$ and
$\operatorname{span}_{K}\\{C_{j}\mid j\in I\\}$ denotes the vector subspace
generated by $\\{C_{j}\mid j\in I\\}$.

Rado’s Theorem for linear homogeneous systems, proven by Rado in [16], reads
as follows:

######  Homogeneous Rado’s Theorem.

Given an $m\times n$ matrix $A$ with rational entries, the system
$A\left(x_{1},\dots,x_{n}\right)=\boldsymbol{0}$ is partition regular over
$\mathbb{N}$ if and only if $A$ satisfies the columns condition.

In particular, for a single equation, Rado’s Theorem tells that the equation
$c_{1}x_{1}+\dots+c_{n}x_{n}=0$ is partition regular over $\mathbb{N}$ if and
only if the following condition (known as Rado’s Condition) holds: there
exists a nonempty set $J\subseteq\\{1,\dots,n\\}$ such that $\sum_{j\in
J}c_{j}=0$.

Whilst there are plenty of results regarding various aspects of partition
regularity of finite and infinite systems of linear equations in the
literature, progress on the general case of functional equations has been
scarce and mostly (but not exclusively) concentrated in the past few years.
For nonlinear polynomial equations we refer to the introductory section of [6]
for a complete list of nonlinear results proven until 2018, and to [7, 1, 8,
14, 15] for the latest advancements we are aware of.

Rado’s Theorem for non-homogeneous linear equations, again proven in [16],
reads as follows:

######  Non-homogeneous Rado’s Theorem.

Let $A$ be an $m\times n$ matrix with rational entries and
$\boldsymbol{b}\in\mathbb{Z}^{m}$. Then the system
$A(x_{1},\dots,x_{n})=\boldsymbol{b}$ is PR over $\mathbb{N}$ if and only if
either

  1. (1)

There is an $a\in\mathbb{N}$ such that $A(a,\dots,a)=\boldsymbol{b}$; or

  2. (2)

The matrix $A$ satisfies the columns condition and there exists
$a\in\mathbb{Z}$ such that $A(a,\dots,a)=\boldsymbol{0}$.

######  Definition 1.4.

Under the notations of Definition (1.1), we say that the system
$\sigma(x_{1},\dots,x_{n})=\boldsymbol{s}$ admits a constant solution if one
can find an $a\in A$ such that $\sigma(a,\dots,a)=\boldsymbol{s}$.

In particular, for a single equation $c_{1}x_{1}+\cdots+c_{n}x_{n}=b$, Rado’s
Theorem states that this equation is PR over $\mathbb{N}$ iff either it has a
constant solution in $\mathbb{N}$, or it has a constant solution in
$\mathbb{Z}$ and there is a non-empty $J\subseteq[n]$ such that $\sum_{i\in
J}c_{i}=0$.

It is easy to see that if a system
$\sigma\left(x_{1},\dots,x_{n}\right)=\boldsymbol{s}$ admits a constant
solution, then it is PR, as constant solutions are obviously
monochromatic222In fact, in many papers only strengthened notions of partition
regularity are considered where only nonconstant solutions or even more
restrictive limitations are imposed. See e.g. [6].. In [9], N. Hindman and I.
Leader proved conditions under which linear systems are non-trivially PR, i.e.
PR linear systems that admit solutions other than constants. In particular,
when $n\geq 3$ they proved that a linear equation
$c_{1}x_{1}+\dots+c_{n}x_{n}=0$ is PR if and only if it is non-trivially
PR333Actually, something strong holds: for $n\geq 3$, one has that in any
finite coloring of $\mathbb{N}$ there exist monochromatic injective solutions
of the equation, namely solutions where all the entries are mutually distinct.
When $n=2$ of course this cannot happen as, by Rado’s Theorem, the only PR
linear equation in two variables is $x-y=0$. We will show in Section 3 that
the same remains true if we substitute ”linear” with ”irreducible”.. By
contrast, our main result shows that whenever the solutions of a system
$\sigma(x_{1},\dots,x_{n})=\boldsymbol{s}$ can be parameterized in function of
one of the variables444Henceforth assumed without loss of generality to be
$(n+1)$-th variable., to be PR is equivalent to admit a constant solution.

######  Main Theorem 1.5.

Let $S$ be an infinite set, $m,n\in\mathbb{N}$,
$\boldsymbol{s}=(s_{1},\dots,s_{m})\in S^{m}$, and
$f_{1},\dots,f_{m}:S^{n+1}\to S$. Let
$\sigma(x_{1},\dots,x_{n},x_{n+1})=\boldsymbol{s}$ be a system as in (1.1).
Suppose that there exists $k\in\mathbb{N}$ such that for all $s\in S$ the
number of solutions in the variables $x_{1},\dots,x_{n}$ of the system
$\sigma\left(x_{1},\dots,x_{n},s\right)=\boldsymbol{s}$ is at most $k$. Then,
the following are equivalent:

  1. (1)

The system is PR over $S$;

  2. (2)

The set of witnesses of the PR of the system is non-empty and for all
witnesses $\mathcal{U}$ of the PR of the system there exists a
$B\in\mathcal{U}$ such that the only solutions of the system in $B$ are
constant; and

  3. (3)

There is a coloring $c_{0}$ of $S$ such that: (a) the system has a
$c_{0}$-monochromatic solution, and (b) all $c_{0}$-monochromatic solutions
the system are constant.

Although we will apply our result to some explicit classes of nonlinear or
exponential polynomials, we notice that Theorem 1.5 is purely set-theoretical:
it holds for arbitrary systems of functional equations on infinite sets.
Theorem 1.5 is proven in Section 2 and its applications are listed in Section
3, which we briefly discuss below.

The first application, see Theorem 3.1, is an extension of Rado’s Theorem for
the PR of systems of Diophantine equations in two variables over $\mathbb{N}$,
thus closing the problem for two variables. Summarizing, to the best of our
knowledge, the only nonlinear classes whose partition regularity over
$\mathbb{N}$ has been completely characterized are555The situation is slightly
different when it comes to the partition regularity over
$\mathbb{R},\mathbb{C}$; see e.g. [13] for a discussion of this fact.,666The
partition regularity of the cases discussed below disconsider constant
solutions, i.e. these characterizations ensure the existence of non-constant
solutions.:

  1. (1)

equations of the form $\sum_{i=1}^{n}c_{i}x_{i}=P(y)$, for $P(y)$ nonlinear
polynomial with no constant term (see [6, Corollary 3.14]); these equations
are partition regular if and only if the linear part satisfies Rado’s
condition, namely if and only if there exists a nonempty $I\subseteq[n]$ such
that $\sum_{i\in I}c_{i}=0$;

  2. (2)

equations of the form $\sum_{i=1}^{h(n)}c_{i}x_{i}^{n}=0$ for $h(n)$ large
enough (namely, $h(n)\geq(1+o(n))n\log n$, see [4] for details), which are
partition regular over $\mathbb{N}$ if and only if there exists a nonempty
$I\subseteq[n]$ such that $\sum_{i\in I}c_{i}=0$;

  3. (3)

equations of the form $\sum_{i=1}^{s}a_{i}x{i}^{2}=\sum_{j=1}^{t}b_{j}y_{j}$,
where the $a_{i},b_{j}$ are non-zero integers, except those of the form
$a_{1}\left(x_{1}^{2}-x_{2}^{2}\right)=a_{2}x_{3}^{2}+b_{1}y_{1}$, for which
the partition regularity is still unknown (see [15, Theorem 1.10]). These
equations are partition regular if and only if there exists $I\neq\emptyset$
such that $\sum_{i\in I}a_{i}=0$ or $\sum_{j\in I}b_{j}=0$.

Rado’s Theorem is known to hold in different settings such as subrings of
$\mathbb{C}$ (see [17]), several families of commutative rings (see [2]) and
for Abelian groups (see [5]); recently, it has also been proven for infinite
integral domains (see [3]). Our second application, see Corollary 3.4,
concerns the partition regularity of linear equations over finite rank
multiplicative subgroups of $\mathbb{C}^{\times}$. We observe an interesting
phenomenon in this setting: the failure of Rado’s Theorem for this class of
equations; in particular, we prove that such sets are additive combinatorial
small in the sense that they do not contain solutions of Schur’s Equation or
$3$-terms arithmetic progressions.

Our third application deals with polyexponential equations over algebraic
number fields777I.e. a finite field extension of the field of rational
numbers., see Theorem 3.6. In recent years, starting with the work of A. Sisto
[20] and then extended by [18], a relevant problem in the area has been the
characterization of the partition regularity of polyexponential equations (and
configurations), namely equations obtained by finite compositions of
exponentiation, sum and product. This problem is far from being well
understood, with only a few positive and even less negative results proven,
such as

  1. (1)

The configuration $\\{x,y,x^{y},xy\\}$ is PR (see [18, Theorem 2]). In
particular, from this it follows that the equations $x=y^{z}$ and $xy=z^{w}$
are PR; and

  2. (2)

The configuration $\\{x,y,x^{y},a,b,a+b\\}$ is not PR (see [18, Theorem 6]).

Applying Theorem 1.5, we study two classes of polyexponential equations whose
characters are mutually coprime and characterize their PR in terms of the
existence of constant solutions; precisely, we consider equations of the forms

$P_{1}(\boldsymbol{x})f_{1}(y)\boldsymbol{\alpha}_{1}^{\boldsymbol{x}}+\cdots+P_{m}(\boldsymbol{x})f_{m}(y)\boldsymbol{\alpha}_{m}^{\boldsymbol{x}}=0,$

and

$Q_{1}(\boldsymbol{x},y)\boldsymbol{\alpha}_{1}^{\boldsymbol{x}}+\cdots+Q_{m}(\boldsymbol{x},y)\boldsymbol{\alpha}_{m}^{\boldsymbol{x}}=0,$

where all $P_{i}$’s and $Q_{i}$’s are polynomial over the rational numbers,
$\boldsymbol{\alpha}_{i}=(\alpha_{i1},\dots,\alpha_{in})\in(\mathbb{Z}\setminus\\{0\\})^{n}$
are such that

$\gcd\\{\alpha_{ij}\mid i\in[m]\text{ and }j\in[m]\\}=1,$

and
$\boldsymbol{\alpha}_{i}^{\boldsymbol{x}}=\alpha_{i1}^{x_{1}}\cdots\alpha_{in}^{x_{n}}$.

##  2\. Proof of the main result

In this section, we provide the proof of our main result by means of
ultrafilters. It will be necessary to quantify how much _injectivity_ we can
possibly find in a given solution of a given system; more precisely:

######  Definition 2.1.

[6, Definition 1.4] Under the conditions of Definition 1.1, we say that the
system $\sigma(x_{1},\dots,x_{n})=\boldsymbol{s}$ is PR over $S$ with
injectivity $\geq r$ ($r\in[n]$), if for all colorings $c$ of $S$ one can find
$c$-monochromatic $a_{1},\dots,a_{n}\in S$ such that
$|\\{a_{1},\dots,a_{n}\\}|\geq r$.

As expected, being PR with a certain injectivity is equivalent to having a
witness of the PR with such injectivity:

######  Theorem 2.2.

[6, Proposition 1.8] Under the conditions of Definition 1.1, the system
$\sigma(x_{1},\dots,x_{n})=\boldsymbol{s}$ is PR over $S$ with injectivity
$\geq r$ if and only if there is an ultrafilter $\mathcal{U}\in\beta S$ such
that for all $A\in\mathcal{U}$ one can find a solution $a_{1},\dots,a_{n}\in
A$ of the system and $|\\{a_{1},\dots,a_{n}\\}|\geq r$.

Additionally, we employ a well-known property of ultrafilters related to fixed
points of functions in $\beta S$:

######  Theorem 2.3.

[10, Theorem 3.35] Let $\mathcal{U}\in\beta S$ and let $f:S\rightarrow S$. Let
$\overline{f}:\beta S\rightarrow\beta S$ be the continuous extension of $f$ to
$\beta S$. Then $\overline{f}(\mathcal{U})=\mathcal{U}$ if and only if there
exists $A\in\mathcal{U}$ such that $f(a)=a$ for all $a\in A$.

Before starting the proof of Theorem 1.5 we prefer to isolate the following
general Lemma we will need, which basically states that a conjunction of
properties is PR if only if at least one of the properties is PR.

######  Lemma 2.4.

Let $\mathcal{U}\in\beta S$ and let
$\varphi_{1}\left(x_{1},\dots,x_{n}\right),\dots,\varphi_{k}\left(x_{1},\dots,x_{n}\right)$
be properties on $S$. The following are equivalent:

  1. (1)

Given any $A\in\mathcal{U}$ there exists $a_{1},\dots,a_{n}\in A$ and
$j\in[k]$ such that $\varphi_{j}(a_{1},\dots,a_{n})$ is satisfied; and

  2. (2)

there exists a $j\in[k]$ such that for all $A\in\mathcal{U}$ one can find
$a_{1},\dots,a_{n}\in A$ satisfying $\varphi_{j}(a_{1},\dots,a_{n})$.

###### Proof.

$(1)\Rightarrow(2)$ Assuming the negation of $(2)$, for all $j\in[k]$ one can
find $A_{j}\in\mathcal{U}$ satisfying $\neg\varphi_{j}(a_{1},\dots,a_{n})$
whenever $a_{1},\dots,a_{n}\in A_{j}$; since $A=A_{1}\cap\dots\cap
A_{k}\in\mathcal{U}$ and $\neg\varphi_{j}(a_{1},\dots,a_{n})$ is satisfied for
all $j\in[k]$, we arrive to a contradiction with $(1)$.

The implication $(2)\Rightarrow(1)$ is immediate. ∎

With Theorems 2.2 and 2.3, and Lemma 2.4, we can prove Theorem 1.5.

###### Proof of Theorem 1.5.

(1)$\Rightarrow$(2). By the hypothesis, the following property holds:

> (P0):
>
>
> There are functions $\psi_{1},\dots,\psi_{k}:S\to S^{n}$ such that whenever
> $a_{1},\dots,a_{n},a_{n+1}\in S$ is a solution of the system
> $\sigma(x_{1},\dots,x_{n},x_{n+1})=\boldsymbol{s}$ one can find a $j\in[k]$
> satisfying $\psi_{j}(a_{n+1})=(a_{1},\dots,a_{n})$.

Let thus $\varphi_{j}(x_{1},\dots,x_{n},x_{n+1})$ be the formula

$\sigma(x_{1},\dots,x_{n},x_{n+1})=\boldsymbol{s}\text{ and
}\psi_{j}(x_{n+1})=(x_{1},\dots,x_{n}).$

As $\sigma(x_{1},\dots,x_{n},)=\boldsymbol{s}$ is PR over $S$, it has a
witness $\mathcal{U}\in\beta S$ of its PR. Fix this witness. Given
$A\in\mathcal{U}$, we claim that there are $j\in[k]$ and
$\boldsymbol{a}=(a_{1},\dots,a_{n},a_{n+1})\in A^{n+1}$ such that
$\varphi_{j}(\boldsymbol{a})$ is satisfied; to prove this claim, we proceed by
contradiction: if the contrary happens, there is an $A\in\mathcal{U}$ such
that for all $j\in[k]$ and $\boldsymbol{a}\in A^{n+1}$ the property
$\varphi_{j}(\boldsymbol{a})$ is not satisfied. By property (P0) above, this
means that $A$ cannot contain a solution of the system
$\sigma(x_{1},\dots,x_{n},x_{n+1})=\boldsymbol{s}$ which contradicts the fact
that $\mathcal{U}$ witnesses the PR of such system.

By Lemma 2.4, there is a $j_{0}\in[k]$ such that for all $A\in\mathcal{U}$ the
following property is satisfied:

> (P1):
>
>
> There exists an $\boldsymbol{a}\in A^{n+1}$ satisfying
> $\varphi_{j_{0}}(\boldsymbol{a})$.

Given any $A\in\mathcal{U}$, we claim that the set

$A_{j_{0}}:=\left\\{a\in A:\exists(a_{1},\dots,a_{n})\in
A^{n}\big{(}\varphi_{j_{0}}(a_{1},\dots,a_{n},a)\big{)}\right\\}$

is a member of $\mathcal{U}$. To prove this claim, we proceed by
contradiction: if not, then $A^{\prime}=A\cap(S\setminus
A_{j_{0}})\in\mathcal{U}$. But it is easily noticeable that $A^{\prime}$ does
not satisfy property (P1), which is a contradiction.

For any given $A\in\mathcal{U}$ and $i\in[k]$, define

$A_{i,j_{0}}=\left\\{a\in A:\pi_{i}\circ\psi_{j_{0}}(a)\in A\right\\},$

where $\pi_{i}:S^{n}\to S$ is the projection onto the $i$th coordinate. One
can easily verify that $A_{j_{0}}\subseteq A_{i,j_{0}}$ and thus
$A_{i,j_{0}}\in\mathcal{U}$. This proves that for all $i\in[n]$ one has that
$\pi_{i}\circ\psi_{j_{0}}(\mathcal{U})=\mathcal{U}$; by Theorem 2.3, there is
a $B_{i}\in\mathcal{U}$ such that $\pi_{i}\circ\psi_{j_{0}}$ is the identity
when restricted to $B_{i}$, and thus the set

$B:=\bigcap_{i=1}^{n}(B_{i})_{j_{0}}$

is the searched set. Indeed, if $(b_{1},\dots,b_{n},b_{n+1})\in B^{n+1}$ is a
solution of the system, by property (P1), one has that
$\psi_{j_{0}}(b_{n+1})=(b_{1},\dots,b_{n})$. It thus must be the case that for
any $i\in[n]$ one has that $b_{i}=\pi_{i}\circ\psi_{j_{0}}(b_{n+1})=b_{n+1}$.

(2)$\Rightarrow$ (3). To prove this implication, we proceed by contradiction.
Since $\mathcal{U}$ is a witness of the PR of the system, we know that there
are monochromatic solutions of any coloring $c$ of $S$; suppose that for all
colorings $c$ of $S$ there is a $c$-monochromatic non-constant solution
$a_{1},\dots,a_{n},a_{n+1}\in S$ to the system. This is easily seen as
equivalent to saying that the system
$\sigma(x_{1},\dots,x_{n},x_{n+1})=\boldsymbol{s}$ is PR over $S$ with
injectivity $\geq 2$. By Theorem 2.2, one can find a witness
$\mathcal{U}\in\beta S$ to this fact; but this is a contradiction with the
hypothesis as $\mathcal{U}$ must contain a set $B$ in which all solutions of
the system are constant.

(3)$\Rightarrow$(1). Since all $c_{0}$ monochromatic solutions of the system
are constant (which exists by the hypothesis), the system is PR for trivial
reasons. ∎

As a trivial consequence, we also get the following result.

######  Corollary 2.5.

In the same hypotheses of Theorem 1.5, for all $\mathcal{U}\in\beta S$ the
following are equivalent:

  1. (1)

$\mathcal{U}$ is a witness of the partition regularity of
$\sigma\left(x_{1},\dots,x_{n}\right)=\boldsymbol{s}$;

  2. (2)

$\Delta_{\sigma}\in\mathcal{U}$, where

$\Delta_{\sigma}=\\{a\in S\mid\sigma(a,\dots,a)=\boldsymbol{s}\\}.$

##  3\. Applications

###  3.1. PR of equations in two variables

We prove that a given system of polynomial equations in two variables over
$\mathbb{Z}$ is PR over $\mathbb{N}$ if and only if it admits constant
solutions. In what follows, $\sigma(x,y)=\boldsymbol{0}$ will denote a system
of polynomial equations. In such a situation, the best we can hope for in
terms of PR is to have a stronger property, that is to any given finite
coloring of $\mathbb{N}$ there are infinitely many monochromatic constant
solutions; in this case, we say that $\sigma(x,y)=\boldsymbol{0}$ is
infinitely PR. As we are going to show, the only irreducible infinitely PR
polynomial in $\mathbb{Z}[x,y]$ is $x-y$. Since the only ingredient needed to
show this fact together with Theorem 1.5 is that polynomials in one variable
over $\mathbb{Z}$ have a uniform bound on their number of solutions that only
depends on the degree of the polynomial, the same result holds _mutatis
mutandis_ for any infinite subset of a given infinite integral domain.

Let us first observe that, when the system consists of homogeneous equations,
this result holds trivially as homogeneous polynomials factorize as products
of linear polynomials over $\mathbb{C}$ (or the algebraic closure of the field
of fractions of the integral domain), so we could directly conclude by Rado’s
Theorem for integral domains and the fact that a polynomial is PR if and only
if at least one of its irreducible factors is (see e.g. [12, Theorem 3.7] and
[3, Theorem A]).

However, Theorem 1.5 allows a very simple proof of the general case.

######  Theorem 3.1.

Let $P_{1},\dots,P_{m}\in\mathbb{Z}\left[x,y\right]$ be polynomials having
degree $\geq 1$ and $\sigma(x,y)=\boldsymbol{0}$ the system

($\star$) $\left\\{\begin{matrix}P_{1}\left(x,y\right)=0,\\\ \vdots\\\
P_{m}\left(x,y\right)=0.\end{matrix}\right.$

The following facts are equivalent:

  1. (1)

The system $\sigma(x,y)=\boldsymbol{0}$ has a constant solution;

  2. (2)

The system $\sigma(x,y)=\boldsymbol{0}$ is PR over $\mathbb{N}$.

###### Proof.

Our method allows us to prove a stronger result, namely: let $R$ be any
infinite integral domain, let $S\subseteq R$ an infinite set, and let
$P_{1},\dots,P_{m}$ be polynomials on two variables over $R$; considering the
system ($\star$ ‣ 3.1), we prove that the following are equivalent:

  1. (1)

The system $\sigma(x,y)=\boldsymbol{0}$ has a constant solution;

  2. (2)

The system $\sigma(x,y)=\boldsymbol{0}$ is PR on $S$.

$(1)\Rightarrow(2)$ This is trivial.

$(2)\Rightarrow(1)$ For each $i\in[m]$ and $s\in S$, consider the monovariate
polynomial $\tilde{P}_{i,s}(x)=P_{i}(x,s)$. Let
$k=\max\\{\operatorname{deg}\left(P_{i,s}\right):i\in[m]\\}$. Then each
$P_{i,s}(x)$ has at most $k$ roots over $R$. We can hence conclude the thesis
by a direct application of Theorem 1.5. ∎

The previous result shows that being PR is equivalent to having at least one
constant solution. As an immediate Corollary of Theorem 3.1, we have the
following characterization of being infinitely PR.

######  Corollary 3.2.

Under the same notations and hypotheses of Theorem 3.1, the following are
equivalent:

  1. (1)

The system $\sigma(x,y)=\boldsymbol{s}$ is infinitely PR over $\mathbb{N}$;

  2. (2)

$(x-y)$ divides $P_{1}\left(x,y\right),\dots,P_{m}\left(x,y\right)$.

In particular, $x-y$ is the only irreducible infinitely PR polynomial in two
variables.

###### Proof.

As before, we can actually prove our theorem in the more general setting where
$R$ is any infinite integral domain, $S\subseteq R$ is an infinite set, and
$P_{1},\dots,P_{m}$ are polynomials over $R$.

$(1)\Rightarrow(2)$ Let us first observe that a general system of functional
equations $\sigma(x_{1},\dots,x_{n})=\boldsymbol{s}$ is infinitely PR over $S$
iff there is a free ultrafilter witness888A _principal_ ultrafilter on $S$ is
an ultrafilter of the form $\\{A\subseteq S\mid s\in A\\}$; a _free_
ultrafilter on $S$ is an ultrafilter that is non-principal. In the case where
$S$ is infinite, if $\mathcal{U}$ is free, then all elements of $\mathcal{U}$
are infinite.. By Theorem 3.1 and the fact that $\mathcal{U}$ does not contain
finite sets, the system $\sigma(x,y)=\boldsymbol{s}$ has an infinite amount of
constant solutions in any $A\in\mathcal{U}$. Then we can apply Bézout Theorem:
if $F$ is the algebraic closure of the fields of fractions of $R$, then it
must be the case that $\\{(x,y)\in S^{2}\mid x=y\\}$ intersect the affine
curve determined by the system $\sigma(x,y)=\boldsymbol{s}$ in an infinite
amount of points, which consequently implies that $x-y$ divides each $P_{i}$.

$(2)\Rightarrow(1)$ This is trivial, as it entails that for all $r\in R\
(r,r)$ is a monochromatic solution of $\sigma(x,y)=0$. ∎

Notice that there are plenty of examples of PR polynomials in two variables
that are not infinitely PR: for example, for all $n\in\mathbb{N}$ the equation
$2x-y=n$ is PR, as it has the constant solution $x=y=n$, but it is not
infinitely PR as it does not have an infinite amount of solutions.

###  3.2. PR of S-unit equations

Let $K$ be a field of characteristic $0$. We denote by $K^{\times}$ the
multiplicative group $(K\setminus\\{0\\},\cdot)$. We say that an Abelian
(multiplicatively written) group $\Gamma$ is _finitely generated_ if there are
$r\in\mathbb{N}$ and $e_{1},\dots,e_{r}\in\Gamma$ such that for all
$g\in\Gamma$ one can find $m,n_{1},\dots,n_{k}\in\mathbb{Z}$ satisfying
$x^{m}=e_{1}^{n_{1}}\dots e_{k}^{n_{k}}$. The smallest $r\in\mathbb{N}$ such
that $\Gamma$ is generated by $r$ elements is called the _rank_ of $\Gamma$.
We fix $\Gamma$ to be a multiplicative subgroup of $\mathbb{C}^{\times}$ of
rank $r$. An $S$-unit equation over $\Gamma$ is an equation of the form

$c_{1}x_{1}+\cdots+c_{n}x_{n}=1$

where $c_{1},\dots,c_{n}\in\mathbb{C}^{\times}$ and unknowns
$x_{1},\dots,x_{n}$ are to be searched inside $\Gamma$. The theory of $S$-unit
equations is an active research area in Diophantine number theory and its
properties have been investigated since the beginning of the XX century. We
refer to the book [11] as a general reference to the subject. We show below,
as an application of Theorem 1.5, that Rado’s Theorem fails to produce
monochromatic solutions of linear equations in $3$ variables over $\Gamma$.
For our purpose, we use the following results to allow us to use Theorem 1.5
to study the PR of $S$-unit equations in 3 variables over $\Gamma$:

######  Theorem 3.3.

[19, Theorem 1.1], [11, Corollary 6.4.5] Let
$G\subseteq\mathbb{C}^{\times}\times\mathbb{C}^{\times}$ be a finitely
generated subgroup of rank $r$. Given $a,b\in\mathbb{C}^{\times}$, the
equation

$ax+by=1$

has at most $2^{8(r+2)}$ solutions $(x,y)\in G$.

If $\Gamma$ has rank $r$, then the rank of $\Gamma\times\Gamma$ is $2r$.
Hence, an $S$-unit equation over $\Gamma$ has at most $2^{16(r+1)}$ solutions.
This result yields the following:

######  Corollary 3.4.

Given any $a,b,c\in\mathbb{C}^{\times}$, the equation $ax+by+cz=0$ is PR over
$\Gamma$ if and only if it has constant solutions, namely if and only if
$a+b+c=0$.

###### Proof.

Since $\Gamma$ is a multiplicative subgroup of $\mathbb{C}^{\times}$, given
any $s\in\Gamma$, solving $ax+by+cs=0$ in $\Gamma$ is equivalent to solving
$a^{\prime}x+b^{\prime}y=1$, where $a^{\prime}=-(a/cs)$ and
$b^{\prime}=-(b/cs)$. By Theorem 3.3, $a^{\prime}x+b^{\prime}y=1$ has a
bounded number of solutions $k$ that only depends only on the rank of
$\Gamma$. Hence, for all $s\in\Gamma$, the equation $ax+by+cs=0$ has at most
$k$ solutions, so Theorem 1.5 implies our thesis. ∎

Hence, the linear version of Rado’s Theorem does not hold in any finite rank
multiplicative subgroups of999By contrast, the group version of Rado’s
Theorem, with multiplication as operation, does hold for $\Gamma$, as it holds
in any infinite group. $\Gamma$ of $\mathbb{C}^{\times}$. As a consequence,
for example, $\Gamma$ will never contain arbitrarily long monochromatic
arithmetic progressions; in fact, it cannot even contain monochromatic
arithmetic progressions of length $3$, as these would form a solution of the
linear equation $x+y-2z=0$.

###  3.3. PR of Polyexponential Equations

An algebraic number field is any finite field extension $K$ of $\mathbb{Q}$,
i.e. $K$ as a vector space over $\mathbb{Q}$ has finite algebraic dimension;
let $d$ be the degree of such extension, i.e. the algebraic dimension of $K$
over $\mathbb{Q}$. As always, $K^{\times}$ denotes the multiplicative group
$(K\setminus\\{0\\},\cdot)$. Let $P_{1},\dots,P_{m}\in K[x_{1},\dots,x_{n}]$
be polynomials of degrees $d_{1},\dots,d_{m}$, respectively. Let
$\boldsymbol{\alpha}_{1},\dots,\boldsymbol{\alpha_{m}}\in(K^{\times})^{n}$.

The _exponential polynomial_ on the variables
$\boldsymbol{x}=(x_{1},\dots,x_{n})$ over $\mathbb{Z}$ with coefficients
$P_{1},\dots,P_{m}$ and characters
$\boldsymbol{\alpha}_{1},\dots,\boldsymbol{\alpha}_{m}$ is

($\star$)
$E(\boldsymbol{x})=P_{1}(\boldsymbol{x})\boldsymbol{\alpha}_{1}^{\boldsymbol{x}}+\cdots
P_{m}(\boldsymbol{x})\boldsymbol{\alpha}_{m}^{\boldsymbol{x}},$

where the _exponential monomial_ $\boldsymbol{\alpha}_{i}^{\boldsymbol{x}}$ is
defined as

$\boldsymbol{\alpha}_{i}^{\boldsymbol{x}}=\alpha_{i1}^{x_{1}}\cdots\alpha_{in}^{x_{n}}.$

A solution of such an equation is any $\boldsymbol{a}\in\mathbb{Z}^{n}$ such
that $E(\boldsymbol{a})=0$.

Henceforth we fix the characters
$\boldsymbol{\alpha}_{1},\dots,\boldsymbol{\alpha}_{m}\in\mathbb{Z}^{n}$,
polynomials in $n$ variables $P_{1},\cdots,P_{\in}K[x_{1},\dots,x_{n}]$, and
functions $f_{1},\dots,f_{m}:\mathbb{Z}\to\mathbb{Z}$; in this subsection we
are interested in studying monochromatic solutions of the equation
$E_{f_{1},\dots,f_{m}}(\boldsymbol{x},y)=0$ over $\mathbb{Z}$, where

$E_{f_{1},\dots,f_{m}}(\boldsymbol{x},y)=P_{1}(\boldsymbol{x})f_{1}(y)\alpha_{1}^{\boldsymbol{x}}+\cdots+P_{m}(\boldsymbol{x})f_{m}(y)\alpha_{m}^{\boldsymbol{x}}.$

The basic theory about solutions of $E(\boldsymbol{x})=0$ can be found in
[19]. We recall below some known facts from that article about such equations
that will enable us to apply Theorem 1.5 to study monochromatic solutions of
$E_{f_{1},\dots,f_{k}}(\boldsymbol{x},y)=0$.

Given a partition $\mathcal{P}$ of $[m]$ and $F\in\mathcal{P}$, let

$E_{F}(\boldsymbol{x})=\sum_{k\in
F}P_{k}(\boldsymbol{x})\boldsymbol{\alpha}_{k}^{\boldsymbol{x}}.$

We let $\sigma_{\mathcal{P}}(\boldsymbol{x})=\boldsymbol{0}$ to be the system
consisting of all the equations of the form $E_{F}(\boldsymbol{x})=0$ as $F$
runs through $\mathcal{P}$. Clearly, if $\mathcal{Q}$ is a
refinement101010I.e. for all $F\in\mathcal{Q}$ one can find a
$G\in\mathcal{P}$ satisfying $F\subseteq G$. of the partition $\mathcal{P}$
then any solution of $\sigma_{\mathcal{Q}}(\boldsymbol{x})=\boldsymbol{0}$ is
also a solution of $\sigma_{\mathcal{P}}(\boldsymbol{x})=\boldsymbol{0}$. In
particular, a solution of
$\sigma_{\mathcal{P}}(\boldsymbol{x})=\boldsymbol{0}$ is also a solution of
the original equation $E(\boldsymbol{x})=0$; note also that a solution for the
original equation has to be a solution for
$\sigma_{\mathcal{P}}(\boldsymbol{x})=\boldsymbol{0}$ for some partition
$\mathcal{P}$ of $[m]$. For a given $\mathcal{P}$, let $S(\mathcal{P})$ be the
set of all solutions of $\sigma_{\mathcal{P}}(\boldsymbol{x})=\boldsymbol{0}$
that does not constitute a solution of
$\sigma_{\mathcal{Q}}(\boldsymbol{x})=\boldsymbol{0}$ for any given proper
refinement $\mathcal{Q}$ of $\mathcal{P}$.

Given
$\boldsymbol{\alpha}_{1},\dots,\boldsymbol{\alpha}_{m}\in\mathbb{Z}^{n}$,
$i,j\in[m]$ and a partition $\mathcal{P}$ of $[m]$, let $i\sim_{\mathcal{P}}j$
mean that there is a $F\in\mathcal{P}$ such that $i,j\in F$. Then the
collection $G(\mathcal{P})$ of all $\boldsymbol{z}\in\mathbb{Z}^{n}$
satisfying
$\boldsymbol{\alpha}_{i}^{\boldsymbol{z}}=\boldsymbol{\alpha}_{j}^{\boldsymbol{z}}$
whenever $i\sim_{\mathcal{P}}j$, form an additive subgroup of
$\mathbb{Z}^{n}$.

Define the constants

$A=\sum_{l=1}^{m}\binom{n+d_{l}}{n}\;\;\text{ and }\;\;B=\max(n,A).$

######  Theorem 3.5.

[19, Theorem 1] If $G(\mathcal{P})$ is trivial, then

$|S(\mathcal{P})|<2^{35B^{3}}d^{6B^{2}}.$

In particular, if $G(\mathcal{P})$ is trivial for all partitions of $[m]$ then
the initial equation admits a uniform bound on the number of solutions, namely
$B_{m}\cdot 2^{35B^{3}}d^{6B^{2}}$, where $B_{m}$ is the $m$-th Bell number
(which is equal to the number of partitions of $[m]$), given by the recurrence
relation $B_{0}=1$ and

$B_{k+1}=\sum_{l=0}^{k}\binom{k}{l}B_{l}.$

Also observe that, if all the polynomials $P_{1},\dots,P_{m}$ are constants,
then $B=\max\\{m,n\\}$. For our purposes, the only fact that matters is that,
when $G(\mathcal{P})$ is trivial for all partitions $\mathcal{P}$ of $[m]$,
the amount of solutions is bounded by a constant that only depends on the
number of variables, namely: the degree of $K$, the degree of the polynomials,
and the number of monomials present in the equation. Hence, a direct
consequence of Theorems 1.5 and 3.5 is:

######  Theorem 3.6.

If the group $G(\mathcal{P})$ is trivial for all possible partitions of $[m]$,
then the equation $E_{f_{1},\dots,f_{m}}(\boldsymbol{x},y)=0$ is partition
regular over $\mathbb{Z}$ if and only if it admits constant solutions.

An immediate example of such equations occurs when
$\alpha_{11},\dots,\alpha_{mn}$ are mutually coprimes; in this case we have
that, for each distinct $k,l\in[m]$ and $\boldsymbol{z}\in\mathbb{Z}^{n}$,
$\boldsymbol{\alpha}_{k}^{\boldsymbol{z}}=\boldsymbol{\alpha}_{l}^{\boldsymbol{z}}$
if and only if $\boldsymbol{z}=\boldsymbol{0}$.

######  Corollary 3.7.

Let $(\alpha_{lk})_{l=1,k=1}^{m,n}$ be a collection of mutually coprime
integers, $P_{1},\dots,P_{m}\in\mathbb{Q}[x_{1},\dots,x_{n}]$ and
$f_{1},\dots,f_{m}:\mathbb{Z}\to\mathbb{Z}$. For each $l\in[m]$ let
$\boldsymbol{\alpha}=(\alpha_{l1},\dots,\alpha_{ln})$. For each $i\in[m]$, let
$A_{i}(w)=P_{i}(w,\dots,w)f_{i}(w)$ and $a_{i}=\alpha_{i1}\cdots\alpha_{in}$.
Then the polynomial exponential equation

$P_{1}(\boldsymbol{x})f_{1}(y)\boldsymbol{\alpha}_{1}^{\boldsymbol{x}}+\dots+P_{m}(\boldsymbol{x})f_{m}(y)\boldsymbol{\alpha}_{m}^{\boldsymbol{x}}=0$

is partition regular over $\mathbb{Z}$ if and only if there exists an
$s\in\mathbb{Z}$ such that

$a_{1}^{s}A_{1}(s)+\cdots+a_{m}^{s}A_{m}(s)=0.$

Another class of polyexponential equations we can study is the class of
equations the form

$P_{1}(\boldsymbol{x},y)\boldsymbol{\alpha}_{1}^{\boldsymbol{x}}+\dots+P_{m}(\boldsymbol{x},y)\boldsymbol{\alpha}_{m}^{\boldsymbol{x}}=0$

where $P_{1},\dots,P_{m}\in K[x_{1},\dots,x_{n},y]$. For a fixed list of such
polynomials, characters, and an $s\in\mathbb{Z}$, the equation

($\star\star$)
$P_{1}(\boldsymbol{x},s)\boldsymbol{\alpha}_{1}^{\boldsymbol{x}}+\dots+P_{m}(\boldsymbol{x},s)\boldsymbol{\alpha}_{m}^{\boldsymbol{x}}=0$

is of the form of the Equation ($\star$ ‣ 3.3). Thus, if the group
$G(\mathcal{P})$ is trivial for all partitions of $[m]$, Theorem 3.5
guarantees that the number of solutions of the Equation ($\star\star$ ‣ 3.3)
is bounded uniformly, i.e. independently from the values of $s$. Hence, by
Theorem 1.5, we prove the following:

######  Theorem 3.8.

If the group $G(\mathcal{P})$ is trivial for all possible partitions of $[m]$,
then the equation

$P_{1}(\boldsymbol{x},y)\boldsymbol{\alpha}_{1}^{\boldsymbol{x}}+\dots+P_{m}(\boldsymbol{x},y)\boldsymbol{\alpha}_{m}^{\boldsymbol{x}}=0$

is partition regular over $\mathbb{Z}$ if and only if it admits constant
solutions.

As before, a basic consequence of the above Theorem is when the characters are
coprime:

######  Corollary 3.9.

Let $(\alpha_{lk})_{l=1,k=1}^{m,n}$ be a collection of mutually coprime
integers and $P_{1},\dots,P_{m}\in\mathbb{Q}[x_{1},\dots,x_{n},y]$. For each
$l\in[m]$ let $\boldsymbol{\alpha}=(\alpha_{l1},\dots,\alpha_{ln})$. For each
$i\in[m]$, let $\tilde{P}_{i}(w)=P_{i}(w,\dots,w)$ and
$a_{i}=\alpha_{i1}\cdots\alpha_{in}$. Then the polynomial exponential equation

$P_{1}(\boldsymbol{x},y)\boldsymbol{\alpha}_{1}^{\boldsymbol{x}}+\dots+P_{m}(\boldsymbol{x},y)\boldsymbol{\alpha}_{m}^{\boldsymbol{x}}=0$

is partition regular over $\mathbb{Z}$ if and only if there exists
$s\in\mathbb{Z}$ such that

$a_{1}^{s}\tilde{P}_{1}(s)+\cdots+a_{m}^{s}\tilde{P}_{m}(s)=0.$

######  Example 3.10.

Let $P_{1}(x,y,z)=xy-z+2$, $P_{2}(x,y,z)=x-y+2z+2$, and $P_{3}(x,y,z)=xy-z+3$.
Then the polyexponential equation

$P_{1}(x,y,z)2^{x}3^{y}+P_{2}(x,y,z)5^{x}7^{y}+P_{3}(x,y,z)11^{x}13^{y}=0$

is not PR over $\mathbb{Z}$. Indeed, if the contrary happens, there exists
$s\in\mathbb{Z}$ such that

($\dagger$)
$6^{s}\tilde{P}_{1}(s)+35^{s}\tilde{P}_{2}(s)+143^{s}\tilde{P}_{3}(s)=0.$

Since

$\tilde{P}_{1}(w)=w^{2}-w+2,\;\;\tilde{P}_{2}(w)=2w+2\;\;\text{ and
}\;\;\tilde{P}_{3}(w)=w^{2}-w+3,$

we have that $2$ divides $\tilde{P}_{1}(n)$ and $\tilde{P}_{2}(n)$ for all
$n\in\mathbb{Z}$. Nevertheless, if there is a common prime factor $p$ of
$\tilde{P}_{1}(s),\tilde{P}_{2}(s)$ and $\tilde{P}_{3}(s)$, then we would have

$x^{2}-x+2\equiv x^{2}-x+3\mod p,$

implying that $2\equiv 3\mod p$, which is absurd. Hence,

$d=\gcd(\tilde{P}_{1}(s),\tilde{P}_{2}(s))>1$

does not divide $\tilde{P}_{3}(s)$. By Bézoult’s Lemma, this a contradiction
with the equation ($\dagger$ ‣ 3.10).

### Acknowledgements

L. Luperi Baglini was supported by the project PRIN 2022 ”Logical methods in
combinatorics”, 2022BXH4R5, MIUR (Italian Ministry of University and
Research). The authors thank the anonymous reviewers for their helpful
comments and suggestions.

## References

  * [1] J. M. Barret, M. Lupini, and J. Moreira, On Rado conditions for nonlinear Diophantine equations, European J. Combin., 94 (2021), p. 103277.
  * [2] V. Bergelson, W. Deuber, N. Hindman, and H. Lefmann, Rado’s theorem for commutative rings, J. Comb. Theory Ser. A, 66 (1994), pp. 68–92.
  * [3] J. Byszewski and E. Krawczyk, Rado's theorem for rings and modules, J. of Comb. Theory, Series A, 180 (2021), p. 105402.
  * [4] S. Chow, S. Lindqvist, and S. Prendiville, Rado's criterion over squares and higher powers, J. Eur. Math. Soc., 23 (2021), pp. 1925–1997.
  * [5] W. Deuber, Partition theorems for abelian groups, J. of Comb. Theory Ser. A, 19 (1975), pp. 95–108.
  * [6] M. Di Nasso and L. Luperi Baglini, Ramsey properties of nonlinear Diophantine equations, Adv. Math., 324 (2018), pp. 84–117.
  * [7] M. Di Nasso and M. Riggio, Fermat-Like Equations that are not Partition Regular, Combinatorica, 18 (2018), pp. 1067 – 1078.
  * [8] S. Farhangi and R. Magner, On the partition regularity of $ax+by=cw^{m}z^{n}$, Integers, 23 (2023), p. A18.
  * [9] N. Hindman and I. Leader, Nonconstant monochromatic solutions to systems of linear equations, in Topics in Discrete Mathematics, M. Klazar, J. Kratochvíl, M. Loebl, J. Matoušek, P. Valtr, and R. Thomas, eds., Springer Berlin Heidelberg, Berlin, 2006, pp. 145–154.
  * [10] N. Hindman and D. Strauss, Algebra in the Stone-Čech Compactification, De Gruyter, Berlin, 2011.
  * [11] J. -H. Evertse and K. Győry, Unit Equations in Diophantine Number Theory, Cambridge University Press, Oct. 2015.
  * [12] L. Luperi Baglini, A nonstandard technique in combinatorial number theory, Eur. J. Comb., 48 (2015), pp. 71–80.  Selected Papers of EuroComb’13.
  * [13] L. Luperi Baglini, Partition regularity of polynomial systems near zero, Semigroup Forum, 103 (2021), pp. 191–208.
  * [14] J. Moreira, Monochromatic sums and products in $\mathbb{N}$, Ann. Math., 185 (2017).
  * [15] S. Prendiville, Counting monochromatic solutions to diagonal Diophantine equations, Discrete Anal., 14 (2021).
  * [16] R. Rado, Studien zur kombinatorik, Math. Z., (1933), pp. 242–280.
  * [17] , Note on combinatorial analysis, Proc. Lond. Math. Soc., s2-48 (1945), pp. 122–160.
  * [18] J. Sahasrabudhe, Exponential patterns in arithmetic ramsey theory, Acta Arith., 182 (2018), pp. 13–42.
  * [19] H. P. Schlickewei and W. P. Schmidt, The number of solutions of polynomial-exponential equations, Compos. Math., 120 (2000), pp. 193–225.
  * [20] A. Sisto, Exponential triples, Electron. J. Combin., 18 (2011).