Tropical geometry (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Tropical geometry" in English language version.

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ams.org

Mikhalkin, Grigory (2005). "Enumerative tropical algebraic geometry in R²" (PDF). Journal of the American Mathematical Society. 18 (2): 313–377. arXiv:math/0312530. doi:10.1090/S0894-0347-05-00477-7.
Katz, Eric (2017), "What is Tropical Geometry?" (PDF), Notices of the American Mathematical Society, 64 (4): 380–382, doi:10.1090/noti1507

archives-ouvertes.fr

hal.archives-ouvertes.fr

Pin, Jean-Eric (1998). "Tropical semirings" (PDF). In Gunawardena, J. (ed.). Idempotency. Publications of the Newton Institute. Vol. 11. Cambridge University Press. pp. 50–69. doi:10.1017/CBO9780511662508.004. ISBN 9780511662508.

arxiv.org

Kontsevich, Maxim; Soibelman, Yan (7 November 2000). "Homological mirror symmetry and torus fibrations". arXiv:math/0011041.
Mikhalkin, Grigory (2005). "Enumerative tropical algebraic geometry in R²" (PDF). Journal of the American Mathematical Society. 18 (2): 313–377. arXiv:math/0312530. doi:10.1090/S0894-0347-05-00477-7.
Cartwright, Dustin; Payne, Sam (2012), "Connectivity of tropicalizations", Mathematical Research Letters, 19 (5): 1089–1095, arXiv:1204.6589, Bibcode:2012arXiv1204.6589C, doi:10.4310/MRL.2012.v19.n5.a10, S2CID 51767353
Hladký, Jan; Králʼ, Daniel; Norine, Serguei (1 September 2013). "Rank of divisors on tropical curves". Journal of Combinatorial Theory, Series A. 120 (7): 1521–1538. arXiv:0709.4485. doi:10.1016/j.jcta.2013.05.002. ISSN 0097-3165. S2CID 3045053.
Tabera, Luis Felipe (1 January 2005). "Tropical constructive Pappus' theorem". International Mathematics Research Notices. 2005 (39): 2373–2389. arXiv:math/0409126. doi:10.1155/IMRN.2005.2373. ISSN 1073-7928. S2CID 14250249.
Kerber, Michael; Gathmann, Andreas (1 May 2008). "A Riemann–Roch theorem in tropical geometry". Mathematische Zeitschrift. 259 (1): 217–230. arXiv:math/0612129. doi:10.1007/s00209-007-0222-4. ISSN 1432-1823. S2CID 15239772.
Chan, Melody; Sturmfels, Bernd (2013). "Elliptic curves in honeycomb form". In Brugallé, Erwan (ed.). Algebraic and combinatorial aspects of tropical geometry. Proceedings based on the CIEM workshop on tropical geometry, International Centre for Mathematical Meetings (CIEM), Castro Urdiales, Spain, December 12–16, 2011. Contemporary Mathematics. Vol. 589. Providence, RI: American Mathematical Society. pp. 87–107. arXiv:1203.2356. Bibcode:2012arXiv1203.2356C. ISBN 978-0-8218-9146-9. Zbl 1312.14142.
Krivulin, Nikolai (2014). "Tropical optimization problems". In Leon A. Petrosyan; David W. K. Yeung; Joseph V. Romanovsky (eds.). Advances in Economics and Optimization: Collected Scientific Studies Dedicated to the Memory of L. V. Kantorovich. New York: Nova Science Publishers. pp. 195–214. arXiv:1408.0313. ISBN 978-1-63117-073-7.
Kalinin, N.; Guzmán-Sáenz, A.; Prieto, Y.; Shkolnikov, M.; Kalinina, V.; Lupercio, E. (15 August 2018). "Self-organized criticality and pattern emergence through the lens of tropical geometry". Proceedings of the National Academy of Sciences of the United States of America. 115 (35): E8135 – E8142. arXiv:1806.09153. Bibcode:2018PNAS..115E8135K. doi:10.1073/pnas.1805847115. ISSN 0027-8424. PMC 6126730. PMID 30111541.
Tourkine, Piotr (2017). "Tropical Amplitudes". Annales Henri Poincaré. 18 (6): 2199–2249. arXiv:1309.3551. Bibcode:2017AnHP...18.2199T. doi:10.1007/s00023-017-0560-7.
Arkani-Hamed, Nima; Trnka, Jaroslav (2014). "The Amplituhedron". JHEP. 2014 (10): 030. arXiv:1312.2007. Bibcode:2014JHEP...10..030A. doi:10.1007/JHEP10(2014)030.
Albrychiewicz, Emil; Ellers, Kai-Isaak; Franco Valiente, Andrés; Hořava, Petr (2024). "Tropological sigma models". JHEP. 06 (6): 135. arXiv:2311.00745. Bibcode:2024JHEP...06..135A. doi:10.1007/JHEP06(2024)135.

berkeley.edu

math.berkeley.edu

Speyer, David; Sturmfels, Bernd (2009), "Tropical mathematics" (PDF), Mathematics Magazine, 82 (3): 163–173, doi:10.1080/0025570X.2009.11953615, S2CID 15278805

doi.org

Maslov, Victor (1987). "On a new superposition principle for optimization problems". Russian Mathematical Surveys. 42 (3): 43–54. Bibcode:1987RuMaS..42...43M. doi:10.1070/RM1987v042n03ABEH001439. S2CID 250889913.
Mikhalkin, Grigory (2005). "Enumerative tropical algebraic geometry in R²" (PDF). Journal of the American Mathematical Society. 18 (2): 313–377. arXiv:math/0312530. doi:10.1090/S0894-0347-05-00477-7.
Pin, Jean-Eric (1998). "Tropical semirings" (PDF). In Gunawardena, J. (ed.). Idempotency. Publications of the Newton Institute. Vol. 11. Cambridge University Press. pp. 50–69. doi:10.1017/CBO9780511662508.004. ISBN 9780511662508.
Simon, Imre (1988). "Recognizable sets with multiplicities in the tropical semiring". Mathematical Foundations of Computer Science 1988. Lecture Notes in Computer Science. Vol. 324. Berlin/Heidelberg: Springer. pp. 107–120. doi:10.1007/BFb0017135. ISBN 978-3-540-50110-7.
Speyer, David; Sturmfels, Bernd (2009), "Tropical mathematics" (PDF), Mathematics Magazine, 82 (3): 163–173, doi:10.1080/0025570X.2009.11953615, S2CID 15278805
Katz, Eric (2017), "What is Tropical Geometry?" (PDF), Notices of the American Mathematical Society, 64 (4): 380–382, doi:10.1090/noti1507
Cartwright, Dustin; Payne, Sam (2012), "Connectivity of tropicalizations", Mathematical Research Letters, 19 (5): 1089–1095, arXiv:1204.6589, Bibcode:2012arXiv1204.6589C, doi:10.4310/MRL.2012.v19.n5.a10, S2CID 51767353
Hladký, Jan; Králʼ, Daniel; Norine, Serguei (1 September 2013). "Rank of divisors on tropical curves". Journal of Combinatorial Theory, Series A. 120 (7): 1521–1538. arXiv:0709.4485. doi:10.1016/j.jcta.2013.05.002. ISSN 0097-3165. S2CID 3045053.
Tabera, Luis Felipe (1 January 2005). "Tropical constructive Pappus' theorem". International Mathematics Research Notices. 2005 (39): 2373–2389. arXiv:math/0409126. doi:10.1155/IMRN.2005.2373. ISSN 1073-7928. S2CID 14250249.
Kerber, Michael; Gathmann, Andreas (1 May 2008). "A Riemann–Roch theorem in tropical geometry". Mathematische Zeitschrift. 259 (1): 217–230. arXiv:math/0612129. doi:10.1007/s00209-007-0222-4. ISSN 1432-1823. S2CID 15239772.
Shiozawa, Yoshinori (2015). "International trade theory and exotic algebras". Evolutionary and Institutional Economics Review. 12 (1): 177–212. doi:10.1007/s40844-015-0012-3. S2CID 155827635. This is a digest of Y. Shiozawa, "Subtropical Convex Geometry as the Ricardian Theory of International Trade" draft paper.
Kalinin, N.; Guzmán-Sáenz, A.; Prieto, Y.; Shkolnikov, M.; Kalinina, V.; Lupercio, E. (15 August 2018). "Self-organized criticality and pattern emergence through the lens of tropical geometry". Proceedings of the National Academy of Sciences of the United States of America. 115 (35): E8135 – E8142. arXiv:1806.09153. Bibcode:2018PNAS..115E8135K. doi:10.1073/pnas.1805847115. ISSN 0027-8424. PMC 6126730. PMID 30111541.
Tourkine, Piotr (2017). "Tropical Amplitudes". Annales Henri Poincaré. 18 (6): 2199–2249. arXiv:1309.3551. Bibcode:2017AnHP...18.2199T. doi:10.1007/s00023-017-0560-7.
Arkani-Hamed, Nima; Trnka, Jaroslav (2014). "The Amplituhedron". JHEP. 2014 (10): 030. arXiv:1312.2007. Bibcode:2014JHEP...10..030A. doi:10.1007/JHEP10(2014)030.
Albrychiewicz, Emil; Ellers, Kai-Isaak; Franco Valiente, Andrés; Hořava, Petr (2024). "Tropological sigma models". JHEP. 06 (6): 135. arXiv:2311.00745. Bibcode:2024JHEP...06..135A. doi:10.1007/JHEP06(2024)135.

harvard.edu

ui.adsabs.harvard.edu

Maslov, Victor (1987). "On a new superposition principle for optimization problems". Russian Mathematical Surveys. 42 (3): 43–54. Bibcode:1987RuMaS..42...43M. doi:10.1070/RM1987v042n03ABEH001439. S2CID 250889913.
Cartwright, Dustin; Payne, Sam (2012), "Connectivity of tropicalizations", Mathematical Research Letters, 19 (5): 1089–1095, arXiv:1204.6589, Bibcode:2012arXiv1204.6589C, doi:10.4310/MRL.2012.v19.n5.a10, S2CID 51767353
Chan, Melody; Sturmfels, Bernd (2013). "Elliptic curves in honeycomb form". In Brugallé, Erwan (ed.). Algebraic and combinatorial aspects of tropical geometry. Proceedings based on the CIEM workshop on tropical geometry, International Centre for Mathematical Meetings (CIEM), Castro Urdiales, Spain, December 12–16, 2011. Contemporary Mathematics. Vol. 589. Providence, RI: American Mathematical Society. pp. 87–107. arXiv:1203.2356. Bibcode:2012arXiv1203.2356C. ISBN 978-0-8218-9146-9. Zbl 1312.14142.
Kalinin, N.; Guzmán-Sáenz, A.; Prieto, Y.; Shkolnikov, M.; Kalinina, V.; Lupercio, E. (15 August 2018). "Self-organized criticality and pattern emergence through the lens of tropical geometry". Proceedings of the National Academy of Sciences of the United States of America. 115 (35): E8135 – E8142. arXiv:1806.09153. Bibcode:2018PNAS..115E8135K. doi:10.1073/pnas.1805847115. ISSN 0027-8424. PMC 6126730. PMID 30111541.
Tourkine, Piotr (2017). "Tropical Amplitudes". Annales Henri Poincaré. 18 (6): 2199–2249. arXiv:1309.3551. Bibcode:2017AnHP...18.2199T. doi:10.1007/s00023-017-0560-7.
Arkani-Hamed, Nima; Trnka, Jaroslav (2014). "The Amplituhedron". JHEP. 2014 (10): 030. arXiv:1312.2007. Bibcode:2014JHEP...10..030A. doi:10.1007/JHEP10(2014)030.
Albrychiewicz, Emil; Ellers, Kai-Isaak; Franco Valiente, Andrés; Hořava, Petr (2024). "Tropological sigma models". JHEP. 06 (6): 135. arXiv:2311.00745. Bibcode:2024JHEP...06..135A. doi:10.1007/JHEP06(2024)135.

mlr.press

proceedings.mlr.press

Zhang, Liwen; Naitzat, Gregory; Lim, Lek-Heng (2018). "Tropical Geometry of Deep Neural Networks". Proceedings of the 35th International Conference on Machine Learning. 35th International Conference on Machine Learning. pp. 5824–5832.

nih.gov

ncbi.nlm.nih.gov

Kalinin, N.; Guzmán-Sáenz, A.; Prieto, Y.; Shkolnikov, M.; Kalinina, V.; Lupercio, E. (15 August 2018). "Self-organized criticality and pattern emergence through the lens of tropical geometry". Proceedings of the National Academy of Sciences of the United States of America. 115 (35): E8135 – E8142. arXiv:1806.09153. Bibcode:2018PNAS..115E8135K. doi:10.1073/pnas.1805847115. ISSN 0027-8424. PMC 6126730. PMID 30111541.

pubmed.ncbi.nlm.nih.gov

Kalinin, N.; Guzmán-Sáenz, A.; Prieto, Y.; Shkolnikov, M.; Kalinina, V.; Lupercio, E. (15 August 2018). "Self-organized criticality and pattern emergence through the lens of tropical geometry". Proceedings of the National Academy of Sciences of the United States of America. 115 (35): E8135 – E8142. arXiv:1806.09153. Bibcode:2018PNAS..115E8135K. doi:10.1073/pnas.1805847115. ISSN 0027-8424. PMC 6126730. PMID 30111541.

ox.ac.uk

economics.ox.ac.uk

"How geometry came to the rescue during the banking crisis". Department of Economics, University of Oxford. Retrieved 24 March 2014.

quantamagazine.org

Hartnett, Kevin (5 September 2018). "Tinkertoy Models Produce New Geometric Insights". Quanta Magazine. Retrieved 12 December 2018.

researchgate.net

Shiozawa, Yoshinori (2015). "International trade theory and exotic algebras". Evolutionary and Institutional Economics Review. 12 (1): 177–212. doi:10.1007/s40844-015-0012-3. S2CID 155827635. This is a digest of Y. Shiozawa, "Subtropical Convex Geometry as the Ricardian Theory of International Trade" draft paper.

semanticscholar.org

api.semanticscholar.org

Maslov, Victor (1987). "On a new superposition principle for optimization problems". Russian Mathematical Surveys. 42 (3): 43–54. Bibcode:1987RuMaS..42...43M. doi:10.1070/RM1987v042n03ABEH001439. S2CID 250889913.
Speyer, David; Sturmfels, Bernd (2009), "Tropical mathematics" (PDF), Mathematics Magazine, 82 (3): 163–173, doi:10.1080/0025570X.2009.11953615, S2CID 15278805
Cartwright, Dustin; Payne, Sam (2012), "Connectivity of tropicalizations", Mathematical Research Letters, 19 (5): 1089–1095, arXiv:1204.6589, Bibcode:2012arXiv1204.6589C, doi:10.4310/MRL.2012.v19.n5.a10, S2CID 51767353
Hladký, Jan; Králʼ, Daniel; Norine, Serguei (1 September 2013). "Rank of divisors on tropical curves". Journal of Combinatorial Theory, Series A. 120 (7): 1521–1538. arXiv:0709.4485. doi:10.1016/j.jcta.2013.05.002. ISSN 0097-3165. S2CID 3045053.
Tabera, Luis Felipe (1 January 2005). "Tropical constructive Pappus' theorem". International Mathematics Research Notices. 2005 (39): 2373–2389. arXiv:math/0409126. doi:10.1155/IMRN.2005.2373. ISSN 1073-7928. S2CID 14250249.
Kerber, Michael; Gathmann, Andreas (1 May 2008). "A Riemann–Roch theorem in tropical geometry". Mathematische Zeitschrift. 259 (1): 217–230. arXiv:math/0612129. doi:10.1007/s00209-007-0222-4. ISSN 1432-1823. S2CID 15239772.
Shiozawa, Yoshinori (2015). "International trade theory and exotic algebras". Evolutionary and Institutional Economics Review. 12 (1): 177–212. doi:10.1007/s40844-015-0012-3. S2CID 155827635. This is a digest of Y. Shiozawa, "Subtropical Convex Geometry as the Ricardian Theory of International Trade" draft paper.

worldcat.org

search.worldcat.org

Hladký, Jan; Králʼ, Daniel; Norine, Serguei (1 September 2013). "Rank of divisors on tropical curves". Journal of Combinatorial Theory, Series A. 120 (7): 1521–1538. arXiv:0709.4485. doi:10.1016/j.jcta.2013.05.002. ISSN 0097-3165. S2CID 3045053.
Tabera, Luis Felipe (1 January 2005). "Tropical constructive Pappus' theorem". International Mathematics Research Notices. 2005 (39): 2373–2389. arXiv:math/0409126. doi:10.1155/IMRN.2005.2373. ISSN 1073-7928. S2CID 14250249.
Kerber, Michael; Gathmann, Andreas (1 May 2008). "A Riemann–Roch theorem in tropical geometry". Mathematische Zeitschrift. 259 (1): 217–230. arXiv:math/0612129. doi:10.1007/s00209-007-0222-4. ISSN 1432-1823. S2CID 15239772.
Kalinin, N.; Guzmán-Sáenz, A.; Prieto, Y.; Shkolnikov, M.; Kalinina, V.; Lupercio, E. (15 August 2018). "Self-organized criticality and pattern emergence through the lens of tropical geometry". Proceedings of the National Academy of Sciences of the United States of America. 115 (35): E8135 – E8142. arXiv:1806.09153. Bibcode:2018PNAS..115E8135K. doi:10.1073/pnas.1805847115. ISSN 0027-8424. PMC 6126730. PMID 30111541.

zbmath.org

Mikhalkin, Grigory (2004). "Amoebas of algebraic varieties and tropical geometry". In Donaldson, Simon; Eliashberg, Yakov; Gromov, Mikhael (eds.). Different faces of geometry. International Mathematical Series. Vol. 3. New York, NY: Kluwer Academic/Plenum Publishers. pp. 257–300. ISBN 978-0-306-48657-9. Zbl 1072.14013.
Chan, Melody; Sturmfels, Bernd (2013). "Elliptic curves in honeycomb form". In Brugallé, Erwan (ed.). Algebraic and combinatorial aspects of tropical geometry. Proceedings based on the CIEM workshop on tropical geometry, International Centre for Mathematical Meetings (CIEM), Castro Urdiales, Spain, December 12–16, 2011. Contemporary Mathematics. Vol. 589. Providence, RI: American Mathematical Society. pp. 87–107. arXiv:1203.2356. Bibcode:2012arXiv1203.2356C. ISBN 978-0-8218-9146-9. Zbl 1312.14142.