A second example of non-Keller mapping
Journal of Applied Mathematics and Computational Mechanics
A SECOND EXAMPLE OF NON-KELLER MAPPING
Sylwia Lara-Dziembek 1, Grzegorz Biernat 1, Edyta Pawlak 1 Magdalena Woźniakowska 2
Institute of Mathematics, Czestochowa University of Technology
2 Faculty of Mathematics and Computer Science, University of Lodz
firstname.lastname@example.org, email@example.com, firstname.lastname@example.org email@example.com
Abstract. In the article the next nontrivial example of non-Keller mapping having two zeros at infinity is analyzed. The rare mapping of two complex variables having two zeros at infinity is considered. In the article it has been proved that if the Jacobian of the considered mapping is constant, then it is zero.
Keywords: Jacobian, zeros at infinity, rare mappings, Keller mapping
In this article we analyze the rare polynomial mappings of two complex variables. We consider the mappings having two zeros at infinity [1-3]. It has been shown that if the Jacobian of such mappings is constant, it must be zero. The work is related to the Keller mapping [4-6] (the Keller mapping is a polynomial mapping with the condition ). In the presented paper, the non-Keller mappings are those for which the Jacobian, if it is constant, is zero.
2. The rare mappings
Let are the complex polynomials of degrees , consequently, and having two zeros at infinity. Assume
where and are the forms of the indicated degrees. These mappings are called rare. Suppose
Let’s prove that .
3. Basic lemma
Let us provide the following property :
Property. If , then .
Lemma. With the given assumptions we have .
Since the Jacobian is constant, we have consecutively
In the 2k-step we have
In the next step we obtain
and taking into account the formula (14), we have
Thus divides (see Property), therefore
In 2k + 2-step we get
Returning to the formulas (18) and (9), we have
Therefore , thus
In the next step we obtain
Back to formula (27) we get
and recalling formula (11) we receive
In the following steps to reduce the power of variables (one with every step). The odd steps are an even power, and even steps are the odd power of the monomial . In the step (3k + 2), the largest power appears, namely . Then, and this means that . Hence (equation (26)), so . Which completes the proof of the lemma.
In the considered example, the form was essential. If we considered the case
where 2k – 2 appears, then difficult and more interesting considerations show that the above case depends on the form . In this paper, the presented case of rare mapping is therefore a “frontier” case, which is rare and non-Keller mapping having two zeros at infinity. Some remarks on the general case
will be presented in the later articles.
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 Pawlak E., Lara-Dziembek S., Biernat G., Woźniakowska M., An example of non-Keller mapping, Journal of Applied Mathematics and Computational Mechanics 2016, 15(1), 115-121.