On weighted bidegree of polynomial automorphisms of C2

• Autorzy: Karaś Marek

Identyfikatory

DOI

10.4064/ba220430-21-3

• Języki publikacji: EN

Abstrakty

EN

Let F=(F1,F2):C2→C2 be a polynomial automorphism. It is well known that degF1|degF2 or degF2|degF1. On the other hand, if (d1,d2)∈N2+=(N∖{0})2 is such that d1|d2 or d2|d1, then one can construct a polynomial automorphism F=(F1,F2) of C2 with degF1=d1 and degF2=d2. Let us fix w=(w1,w2)∈N2+ and consider the weighted degree on C[x,y] with degwx=w1 and degwy=w2. In this note we address the structure of the set {(degwF1,degwF2):(F1,F2) is an automorphism of C2}. This is a very first, but necessary, step in studying weighted multidegrees of polynomial automorphisms.

Słowa kluczowe

EN

polynomial automorphism; tame automorphism; wild automorphism; multidegree; weighted multidegree

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2022
• Tom: Vol. 70, no. 2
• Strony: 107--114
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Karaś Marek

• Faculty of Applied Mathematics, AGH University of Science and Technology, 30-059 Kraków, Poland

• https://orcid.org/0000-0003-0821-521X

Bibliografia

• [1] A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture, Birkhäuser, Basel, 2000.
• [2] E. Edo, T. Kanehira, M. Karaś and S. Kuroda, Separability of wild automorphisms of a polynomial ring, Transform. Groups 18 (2013), 81-96.
• [3] J.-P. Furter, On the variety of automorphisms of the affine plane, J. Algebra 195 (1997), 604-623.
• [4] H. W. E. Jung, Über ganze birationale Transformationen der Ebene, J. Reine Angew. Math. 184 (1942), 161-174.
• [5] M. Karaś, There is no tame automorphism of C3 with multidegree (3, 4, 5), Proc. Amer. Math. Soc. 139 (2011) 769-775.
• [6] M. Karaś, Tame automorphisms of C3 with multidegree of the form (3, d2, d3), J. Pure Appl. Algebra 214 (2010), 2144-2147.
• [7] M. Karaś, Multidegrees of tame automorphisms of Cn, Dissertationes Math. 477 (2011), 55 pp.
• [8] M. Karaś and J. Zygadło, On multidegree of tame and wild automorphisms of C3, J. Pure Appl. Algebra 215 (2011), 2843-2846.
• [9] W. van der Kulk, On polynomial rings in two variables, Nieuw Arch. Wisk. (3) 1 (1953), 33-41.
• [10] S. Kuroda, On the Karaś type theorems for the multidegrees of polynomial automorphisms, J. Algebra 423 (2015), 441-465.
• [11] S. Kuroda, Weighted multidegrees of polynomial automorphisms over a domain, J. Math. Soc. Japan 68 (2016), 119-149.
• [12] X. Sun and Y. Chen, Multidegrees of tame automorphisms in dimension three, Publ. RIMS Kyoto Univ. 48 (2012), 129-137.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).