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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.

Complexity of the Inversion Algorithm of Polynomial Mappings

Bogdan P.

Schedae Informaticae

2016

Vol. 25

209--225

In this paper we will recall the inversion algorithm described in [1]. The algorithm classifies polynomial automorphisms into two sets: Pascal finite and Pascal infinite. In this article the complexity of the inversion algorithm will be estimated. To do so, we will present two popular ways how Computer Algebra Systems (CASes) keep the information about multivariate polynomials. We will define the complexity as the amount of simple operations performed by the algorithm as a function of the size of the input. We will define simple operations of the algorithm. Then we will estimate complexity of checking that the polynomial map is not a polynomial automorphism. To do so we will use theorem 3.1 from [1].

Wild Multidegrees of the Form (d,d2,d3) for Fixed d≥3

Karaś M., Zygadło J.

Bulletin of the Polish Academy of Sciences. Mathematics

2012

Vol. 60, no 3

211--218

Let d be any integer greater than or equal to 3. We show that the intersection of the set mdeg(Aut(C3))∖mdeg(Tame(C3)) with {(d1,d2,d3)∈(N+)3:d=d1≤d2≤d3} has infinitely many elements, where mdegh=(degh1,…,deghn) denotes the multidegree of a polynomial mapping h=(h1,…,hn):Cn→Cn. In other words, we show that there are infinitely many wild multidegrees of the form (d,d2,d3), with fixed d≥3 and d≤d2≤d3, where a sequence (d1,…,dn)∈Nn is a wild multidegree if there is a polynomial automorphism F of Cn with mdegF=(d1,…,dn), and there is no tame automorphism of Cn with the same multidegree.

Tame automorphisms of C3 with multidegree of the form (p1, p2, d3)

Karaś M.

Bulletin of the Polish Academy of Sciences. Mathematics

2011

Vol. 59, no 1

27-32

Let d3 ≥ p2 > p1 ≥ 3 be integers such that p1, p2 are prime numbers. We show that the sequence (p1, p2, d3) is the multidegree of some tame automorphism of C3 if and only if d3 ∈ p1N p2N, i.e. if and only if d3 is a linear combination of p1 and p2 with coefficients in N.