A Galois structure on the orbit of large steps walks in the quadrant
The enumeration of weighted walks in the quarter plane reduces to studying a functional equation with two catalytic variables. When the steps of the walk are small, Bousquet-Mélou and Mishna defined a group called the group of the walk which turned out to be crucial in the classification of the smal...
Saved in:
Published in | arXiv.org |
---|---|
Main Authors | , |
Format | Paper |
Language | English |
Published |
Ithaca
Cornell University Library, arXiv.org
17.09.2024
|
Subjects | |
Online Access | Get full text |
Cover
Loading…
Abstract | The enumeration of weighted walks in the quarter plane reduces to studying a functional equation with two catalytic variables. When the steps of the walk are small, Bousquet-Mélou and Mishna defined a group called the group of the walk which turned out to be crucial in the classification of the small steps models. In particular, its action on the catalytic variables provides a convenient set of changes of variables in the functional equation. This particular set called the orbit has been generalized to models with arbitrary large steps by Bostan, Bousquet-Mélou and Melczer (BBMM). However, the orbit had till now no underlying group. In this article, we endow the orbit with the action of a Galois group, which extends the notion of the group of the walk to models with large steps. As an application, we look into a general strategy to prove the algebraicity of models with small backwards steps, which uses the fundamental objects that are invariants and decoupling. The group action on the orbit allows us to develop a Galoisian approach to these two notions. Up to the knowledge of the finiteness of the orbit, this gives systematic procedures to test their existence and construct them. Our constructions lead to the first proofs of algebraicity of weighted models with large steps, proving in particular a conjecture of BBMM, and allowing to find new algebraic models with large steps. |
---|---|
AbstractList | The enumeration of weighted walks in the quarter plane reduces to studying a functional equation with two catalytic variables. When the steps of the walk are small, Bousquet-Mélou and Mishna defined a group called the group of the walk which turned out to be crucial in the classification of the small steps models. In particular, its action on the catalytic variables provides a convenient set of changes of variables in the functional equation. This particular set called the orbit has been generalized to models with arbitrary large steps by Bostan, Bousquet-Mélou and Melczer (BBMM). However, the orbit had till now no underlying group. In this article, we endow the orbit with the action of a Galois group, which extends the notion of the group of the walk to models with large steps. As an application, we look into a general strategy to prove the algebraicity of models with small backwards steps, which uses the fundamental objects that are invariants and decoupling. The group action on the orbit allows us to develop a Galoisian approach to these two notions. Up to the knowledge of the finiteness of the orbit, this gives systematic procedures to test their existence and construct them. Our constructions lead to the first proofs of algebraicity of weighted models with large steps, proving in particular a conjecture of BBMM, and allowing to find new algebraic models with large steps. |
Author | Bonnet, Pierre Hardouin, Charlotte |
Author_xml | – sequence: 1 givenname: Pierre surname: Bonnet fullname: Bonnet, Pierre – sequence: 2 givenname: Charlotte surname: Hardouin fullname: Hardouin, Charlotte |
BookMark | eNqNyk0OgjAQQOHGaCIqd5jENUlpLbg1_h7APSlaFGxamGnj9SXRA7h6i-8t2NR5ZyYsEVLm2XYjxJylRB3nXBSlUEom7LCDs7a-JaCA8RYiGvAOwnMM1m0A34DV-DCjm57gre2LoP0eQ9R31C6s2KzRlkz665KtT8fr_pL16IdoKFSdj-hGqmTOC6VEWXL53_UBSA87xw |
ContentType | Paper |
Copyright | 2024. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
Copyright_xml | – notice: 2024. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
DBID | 8FE 8FG ABJCF ABUWG AFKRA AZQEC BENPR BGLVJ CCPQU DWQXO HCIFZ L6V M7S PIMPY PQEST PQQKQ PQUKI PRINS PTHSS |
DatabaseName | ProQuest SciTech Collection ProQuest Technology Collection Materials Science & Engineering Collection ProQuest Central (Alumni) ProQuest Central ProQuest Central Essentials ProQuest Central Technology Collection ProQuest One Community College ProQuest Central SciTech Premium Collection ProQuest Engineering Collection Engineering Database Publicly Available Content Database ProQuest One Academic Eastern Edition (DO NOT USE) ProQuest One Academic ProQuest One Academic UKI Edition ProQuest Central China Engineering Collection |
DatabaseTitle | Publicly Available Content Database Engineering Database Technology Collection ProQuest Central Essentials ProQuest One Academic Eastern Edition ProQuest Central (Alumni Edition) SciTech Premium Collection ProQuest One Community College ProQuest Technology Collection ProQuest SciTech Collection ProQuest Central China ProQuest Central ProQuest Engineering Collection ProQuest One Academic UKI Edition ProQuest Central Korea Materials Science & Engineering Collection ProQuest One Academic Engineering Collection |
DatabaseTitleList | Publicly Available Content Database |
Database_xml | – sequence: 1 dbid: 8FG name: ProQuest Technology Collection url: https://search.proquest.com/technologycollection1 sourceTypes: Aggregation Database |
DeliveryMethod | fulltext_linktorsrc |
Discipline | Physics |
EISSN | 2331-8422 |
Genre | Working Paper/Pre-Print |
GroupedDBID | 8FE 8FG ABJCF ABUWG AFKRA ALMA_UNASSIGNED_HOLDINGS AZQEC BENPR BGLVJ CCPQU DWQXO FRJ HCIFZ L6V M7S M~E PIMPY PQEST PQQKQ PQUKI PRINS PTHSS |
ID | FETCH-proquest_journals_31065527703 |
IEDL.DBID | BENPR |
IngestDate | Thu Oct 10 21:48:19 EDT 2024 |
IsOpenAccess | true |
IsPeerReviewed | false |
IsScholarly | false |
Language | English |
LinkModel | DirectLink |
MergedId | FETCHMERGED-proquest_journals_31065527703 |
OpenAccessLink | https://www.proquest.com/docview/3106552770?pq-origsite=%requestingapplication% |
PQID | 3106552770 |
PQPubID | 2050157 |
ParticipantIDs | proquest_journals_3106552770 |
PublicationCentury | 2000 |
PublicationDate | 20240917 |
PublicationDateYYYYMMDD | 2024-09-17 |
PublicationDate_xml | – month: 09 year: 2024 text: 20240917 day: 17 |
PublicationDecade | 2020 |
PublicationPlace | Ithaca |
PublicationPlace_xml | – name: Ithaca |
PublicationTitle | arXiv.org |
PublicationYear | 2024 |
Publisher | Cornell University Library, arXiv.org |
Publisher_xml | – name: Cornell University Library, arXiv.org |
SSID | ssj0002672553 |
Score | 3.5708625 |
SecondaryResourceType | preprint |
Snippet | The enumeration of weighted walks in the quarter plane reduces to studying a functional equation with two catalytic variables. When the steps of the walk are... |
SourceID | proquest |
SourceType | Aggregation Database |
SubjectTerms | Decoupling Enumeration Functional equations |
Title | A Galois structure on the orbit of large steps walks in the quadrant |
URI | https://www.proquest.com/docview/3106552770 |
hasFullText | 1 |
inHoldings | 1 |
isFullTextHit | |
isPrint | |
link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwfV3dS8MwED_ciuCbn-icI6CvxTXpmvZJ_Gg3hI0hCnsbSZtCsaxd0-Gbf7uXrtMHYY_hQsKFcL-73x13AHejwEkUo7EtBXMxQHGoLQLK7aFIg1RynkplCP3pzJt8uK-L0aIl3HRbVrmziY2hTorYcOT36IZ4plsYHz6Ua9tMjTLZ1XaERgcs6rgmTWs9hbP52y_LQj2OPjP7Z2gb9IiOwZqLUlUncKBWp3DYFF3G-gxeHslY5EWmybaL66ZSpFgRdMlIUcmsJkVKclOpjXJVavIl8k9Nsu2O9UYkiDP1OdxG4fvzxN5dvWy_h17-KcMuoItxvroE4iSuLwT1Y5pIDCNYoHzmIwJzLzZN1dUV9Ped1NsvvoYjinhsSh0c3ocu6qVuEE9rOYCOH40H7dPhavod_gDtUX93 |
link.rule.ids | 786,790,12792,21416,33406,33777,43633,43838 |
linkProvider | ProQuest |
linkToHtml | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwfV3dS8MwED-0RfTNT_yYGtDX4pp2TfskfmxW3cqQCXsbSZtCsaxd0-G_76Xr9EHY84WEC-F-d7_8uAO47QV2Ih0aW4I7LhYoNrV4QJnV5WmQCsZSITWhP4q88NN9m_amLeGmWlnlOiY2gTopYs2R32Ea4uluYax7Xy4sPTVK_662IzS2wdQtN30DzMd-NP74ZVmoxzBndv4F2gY9BvtgjnkpqwPYkvND2GlEl7E6gucH8sLzIlNk1cV1WUlSzAmmZKSoRFaTIiW5VmqjXZaKfPP8S5FstWKx5AniTH0MN4P-5Cm01kfP2uehZn_OOCdgYJ0vT4HYietzTv2YJgLLCCeQ6BEiMPNi3VRdnkFn007nm83XsBtORsPZ8DV6v4A9itisZQ8264CBPspLxNZaXLUX-AMTs4BW |
openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=A+Galois+structure+on+the+orbit+of+large+steps+walks+in+the+quadrant&rft.jtitle=arXiv.org&rft.au=Bonnet%2C+Pierre&rft.au=Hardouin%2C+Charlotte&rft.date=2024-09-17&rft.pub=Cornell+University+Library%2C+arXiv.org&rft.eissn=2331-8422 |