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2 Bulletin of the Polish Academy of Sciences. Mathematics

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On weighted bidegree of polynomial automorphisms of C2

Karaś Marek

Bulletin of the Polish Academy of Sciences. Mathematics

2022

Vol. 70, no. 2

107--114

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.

A note on the solution set of the equation [Lr1, P1]=[P2, Ls2] for given linear forms L1, L2

Holik Daria, Karaś Marek

Bulletin of the Polish Academy of Sciences. Mathematics

2021

Vol. 69, no. 1

11--20

Let k be a field of characteristic zero. In this note, for given linear forms L1, L2 ∈ k[x1,…, xn] and given r, s ∈ N+ = N \ {0}, we consider the equation [Lr1, P1] =[P2, Ls2] with unknowns P1, P2 ∈ k[x1, . . . , xn], and give a complete description of the set of all solutions of such an equation. Equivalently, the above equation can be written as anequation for differential forms: d(Lr1) ∧ dP1 = dP2 ∧ d(Ls2).