Issue 2&3, Volume 32, August 2007

2007 · Open access

1. $\mathbb Q$-FACTORIAL SUBALGEBRAS OF A POLYNOMIAL RING MASAYOSHI MIYANISHI PDF

   ▸ Abstract

   We give an algebraic characterization of a log affine pseudo-plane and the quotient surface $\mathbb A^2/G$ of $\mathbb A^2$ with a small finite subgroup $G$ of $GL(2,\mathbb C)$ in terms of $\mathbb Q$-factorial normal subalgebras of a polynomial ring $\mathbb C[x, y]$. Then we consider the cancellation problem for these surfaces.

   113-122
2. NOTE ON BOUNDARY OBSTRUCTION TO JACOBIAN CONJECTURE OF TWO VARIABLES MUTSUO OKA PDF

   123-139
3. AN ESTIMATION OF THE NUMBER OF BIFURCATION VALUES FOR REAL POLYNOMIALS HÀ HUY VUI, PHẠM TIẾN SƠN PDF

   ▸ Abstract

   Let $f : \mathbb R^n \to \mathbb R$ be a polynomial of degree $d$ with only isolated complex critical points. It is shown that the set of bifurcation values of $f$ is contained in a set which has at most $(d −1)^n$ points. The proof of this result is done in such a way that all points of the last set can be explicitly calculated. As a consequence, we obtain a finite set containing the global infimum value of a bounded below polynomial.

   141-153
4. SOME FAMILIES OF POLYNOMIAL AUTOMORPHISMS II. ERIC EDO PDF

   ▸ Abstract

   We work with families of polynomial automorphisms of the complex affine plane whose generic length is 2. We obtain two new results which show some differences between the behavior of these families and the family of generic length 3 studied in [2].

   155-168
5. SIMILARITY METHOD FOR THE JACOBIAN PROBLEM SHREERAM S. ABHYANKAR PDF

   ▸ Abstract

   This is an expository article giving a modified version of my talk at the October 2006 Conference in Hanoi.

   169-175
6. THE JACOBIAN VARIETY RONEN PERETZ PDF

   177-187
7. ON THE ŁOJASIEWICZ EXPONENT AT INFINITY OF POLYNOMIAL MAPPINGS TADEUSZ KRASIŃSKI PDF

   ▸ Abstract

   This is a survey article on the Łojasiewicz exponent at infinity of polynomial mappings. This exponent is a numerical measure of behaviour of polynomial mappings at infinity. We present properties, formulas and some applications of the Łojasiewicz exponent at infinity.

   189-203
8. A PROOF OF THE EQUIVALENCE OF THE DIXMIER, JACOBIAN AND POISSON CONJECTURES PASCAL KOSSIVI ADJAMAGBO, ARNO VAN DEN ESSEN PDF

   205-214
9. ON THE METHODS TO CONSTRUCT UFD COUNTEREXAMPLES TO A CANCELLATION PROBLEM STEFAN MAUBACH PDF

   ▸ Abstract

   In a previous paper [3] the author together with prof. dr. Finston constructed a class of UFDs $A_{n,m}$ where $n,m \in \mathbb N$. These rings are all stably equivalent $(A_{n,m}[T] \cong A_{p,q}[T]$ for all $n, m, p, q)$ but are only isomorphic themselves if $(n,m) = (p, q)$. These examples are the first UFD examples over a characteristically closed field satisfying this behavior. In this paper, we describe the methods used in this article, and show that they are very general, enabling the reader to construct many more such examples, based on the same principles.

   215-222
10. ADDITIVE GROUP ACTIONS: QUOTIENTS, INVARIANTS, AND CANCELLATION DAVID R. FINSTON PDF

    223-232
11. HERBIE: A CUBIC LINEAR COUNTEREXAMPLE TO THE DEPENDENCE PROBLEM MICHIEL DE BONDT PDF

    ▸ Abstract

    We show that there exists a cubic linear counterexample to the (Homogeneous) Dependence Problem in dimension 53. More generally, there exists a power linear counterexample of degree $d$ to the (H)DP (in sufficiently large dimensions), if and only if there exists any counterexample of degree $d$ to the (H)DP.

    233-241
12. A NOTE ON A NOTE BY MUTSUO OKA PIERRETTE CASSOU-NOGUÈS PDF

    243-246
13. HILBERT’S FOURTEENTH PROBLEM AND ALGEBRAIC EXTENSIONS WITH AN APPENDIX ON ROBERTS TYPE COUNTEREXAMPLES SHIGERU KURODA PDF

    247-257
14. A VANISHING CONJECTURE ON DIFFERENTIAL OPERATORS WITH CONSTANT COEFFICIENTS WENHUA ZHAO PDF

    ▸ Abstract

    In the recent progress [4], [17] and [25], the well-known JC (Jacobian conjecture) ([2], [10]) has been reduced to a VC (vanishing conjecture) on the Laplace operators and HN (Hessian nilpotent) polynomials (the polynomials whose Hessian matrix are nilpotent). In this paper, we first show that the vanishing conjecture above, hence also the JC, is equivalent to a vanishing conjecture for all 2nd order homogeneous differential operators $\Lambda$ and $\Lambda$-nilpotent polynomials $P$ (the polynomials $P(z)$ satisfying $\Lambda^m P^m = 0$ for all $m \geqslant 1$). We then transform some results in the literature on the JC, HN polynomials and the VC of the Laplace operators to certain results on -nilpotent polynomials and the associated VC for 2nd order homogeneous differential operators $\Lambda$. This part of the paper can also be read as a short survey on HN polynomials and the associated VC in the more general setting. Finally, we discuss a still-to-be-understood connection of $\Lambda$-nilpotent polynomials in general with the classical orthogonal polynomials in one or more variables. This connection provides a conceptual understanding for the isotropic properties of homogeneous $\Lambda$-nilpotent polynomials for 2nd order homogeneous full rank differential operators $\Lambda$ with constant coefficients.

    259-286
15. A NOTE ON SINGULARITY AND NON-PROPER VALUE SET OF POLYNOMIAL MAPS OF $\mathbb C^2$ NGUYEN VAN CHAU PDF

    ▸ Abstract

    Some properties of the relation between the singular point set and the non-proper value curve of polynomial maps of $\mathbb C^2$ are expressed in terms of Newton-Puiseux expansions.

    287-294
16. THE JACOBIAN CONJECTURE FOR RATIONAL POLYNOMIALS LÊ DŨNG TRÁNG PDF

    ▸ Abstract

    In this paper we consider the Jacobian conjecture in dimension 2 over the field $\mathbb C$ of complex numbers, when one of the two polynomial functions has generic rational fibers. We call rational polynomials polynomials whose general fiber is a non-singular rational curve.

    295-301
17. SOME OPEN QUESTIONS ON POLYNOMIAL AUTOMORPHISMS AND RELATED TOPICS TAC KAMBAYASHI PDF

    ▸ Abstract

    This is a collection of open questions presented at the Open Session of the International School and Workshop “Polynomial Automorphisms and Related Topics”, October 9-20, 2006, Hanoi, Vietnam.

    303-318