# The Jacobians of non-maximal degree

### Sylwia Lara-Dziembek

,### Grzegorz Biernat

,### Edyta Pawlak

Journal of Applied Mathematics and Computational Mechanics |
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THE JACOBIANS OF NON-MAXIMAL DEGREE

Sylwia Lara-Dziembek, Grzegorz Biernat, Edyta Pawlak

Institute of Mathematics, Czestochowa
University of Technology

Częstochowa, Poland

sylwia.lara@im.pcz.pl, grzegorz.biernat@im.pcz.pl, edyta.pawlak@im.pcz.pl

Received: 30 October
2016; accepted: 15 November 2016

**Abstract.** In
the article the leading forms of the polynomial mapping having the Jacobians
of non-maximal degree are considered. In particular, the mappings having two
zeros
at infinity are discussed.

*Keywords: **Jacobian, zeros
at infinity*

1. Introduction

The article presents the decomposition of leading forms of the polynomial mapping of two complex variables in the case when the Jacobian of this mapping does not meet maximal degree. In particular, the structure of these forms interest us in the case where the mapping has two zeros at infinity.

2. The Jacobian having non-generic degrees

Let *f _{m}*,

*h*be the forms of variables

_{n }*X*,

*Y*of degrees

*m*,

*n*respectively with the condition . Suppose that the Jacobian of forms

*f*,

_{m}*h*vanish and represent the structures of these forms.

_{n}**Lemma.** *Let **. Therefore*

(1) |

*and*

(2) |

*where:*

*and* , , , where means
the greatest common divisor
of the numbers *m* and *n*.

*Proof.*

According to the Euler formula [1] we have

(3) |

so using the Cramer rule we obtain

(4) |

Because , then

(5) |

so after dividing by the greatest common divisor
(*m*, *n*) of the numbers *m *and *n*
we receive

(6) |

This means that
the logarithmic derivatives of the form *f _{m}*,

*h*respectively, satisfy equalities

_{n }(7) |

therefore

(8) |

Thus the forms *f _{m}*,

*h*have the same factors.

_{n}So

(9) |

and

(10) |

According to formula (9) the exponents satisfy the equalities

(11) |

Since the numbers are relatively prime numbers, then

(12) |

So

(13) |

therefore

(14) |

and at the same time the equality hold.

This means that

(15) |

and

(16) |

Assume

(17) |

(18) |

where .

Therefore

(19) |

so

(20) |

Thus exactly when . However

(21) |

and

(22) |

Then

(23) |

so

(24) |

This means that

(25) |

therefore

(26) |

This completes the proof.

**Remark 1. **Obviously
we can assume that .

**Corollary
1 **[2]**.*** Let** *,* where *.* If *, *then only zeros*
*at infinity* of the mapping (*f*, *h*) *are the factors
of the form f _{m} or h_{n}*.

**Corollary
2.*** If the
numbers m and n are relatively prime and *, *then ** and *. *This means that the mapping
*(*f*, *h*)* has only one zero at infinity.*

**Corollary
3.*** Let** *,* where *.* Let *. *If the mapping *(*f*,
*h*)* have two zeros at infinity, then*

(27) |

*where **, k and l are **relatively prime, *.

**Remark 2.** *In particular, **can be** consider the case* (*we put k* = l = 1, *p* = *k*
+1, *q* = *k*). *The mappings of this type were considered in the
paper *[3].

3. Conclusion

The Jacobians of non-maximal degree appears
for the mappings with the constant Jacobian. The
Jacobians Conjecture
[4-9] do not occur for non trivial classes of the
mappings having the constant Jacobian and one or two zeros in infinity
[3, 10]. In the second case (two zeros in infinity) the leading forms of
the mapping have the form given in the** **Corollary 2. The study of such mappings lead
to
the general classes of the mappings for which the Jacobian Conjecture does not
take place.

References

[1] Mostowski A., Stark M., Elementy algebry wyższej, Wyd. Naukowe PWN, Warszawa 1997.

[2] Biernat G., The Jacobians of Lower Degree, Scientific Research of the Institute of Mathematics and Computer Science 2003, 2(1), 19-24.

[3] Lara-Dziembek S., Biernat G., Pawlak E., Woźniakowska M., A second example of non-Keller mapping, Journal of Applied Mathematics and Computational Mechanics 2016, 15(2), 65-70.

[4] Griffiths P., Harris J., Principles of Algebraic Geometry, New York 1978.

[5] Mumford D., Algebraic Geometry I: Complex Projective Varieties , Springer Verlag, New York 1975.

[6] Shafarevich I.R., Basic Algebraic Geometry, Berlin, Springer-Verlag, New York 1974.

[7] Wright D., On the Jacobian conjecture, no. 3, 423-440, Illinois J. Math. 25, 1981.

[8] Van den Essen A., Polynomial automorphisms and the Jacobian conjecture, Progress in Matematics 190, Birkhäuser Verlag, Basel 2000.

[9] Bass H., Connell E.H., Wright D., The Jacobian conjecture: reduction of degree and formal expansion of the inverse, American Mathematical Society. Bulletin. New Series 1982, 7(2), 287-330.

[10] 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.