Prof. Peretz Ronen
Born: 1958, Israel
Ph.D. 1989, Technion (Faculty of Mathematics)
Senior Lecturer 1995.
 Department of Mathematics - Head

Research Interests:
Classification of polynomial mappings between affine varieties in connection to problems like the Jacobian Conjecture.
Research Projects:
Polynomial mappings and the Jacobian conjecture. Algorithms in commutative algebra. Algorithms in screening, halftoning and digital printing.
Abstracts of Current Research:
• On Counterexamples To Kellers Problem: Let $F:K^{n} \rightarrow K^{n}$ be a polynomial map where $K=R$ or $C$. Let usdenote by$J(F)$ the determinant of the Jacobian of $F$. The Jacobian conjecture is thefollowing statement : \If $J(F)$ never vanishes then the map $F$ is injective. \Originally, the conjecture was stated for $K=C$ with polynomials over $Z$by O. Keller.The conjecture for $K=C$ is still open for $n \geq 2$. \The conjecture for $K=R$, the so called, real Jacobian conjecture was recentlyshown tobe false by S. Pinchuk.The main purpose of this paper is to give a proof to the fact that there is nocounterexampleto the complex Jacobian conjecture of the type constructed by S. Pinchuk for therealcase. \It is explained how can one view Pinchuks construction in terms of theasymptotic values of themap. As a consequence it is proved that there are special kinds of polynomialrings such thatfinding a Jacobian pair within them establishes a counterexample to theconjecture. The simplestsuch a ring is $K[V,VU,VU^{2}+U]$. For $K=R$ Pinchuk found a pair $P,Q \inR[V,VU,VU^{2}+U]$whose Jacobian is always positive. We prove that there is no Jacobian pair in$C[V,VU,VU^{2}+U]$.
• The Geometry of the Asymptotics of Polynomial Mappings: The aim of this paper is to develop a theory for the asymptotic behavior of polynomials and of polynomial maps over R and over C and to apply it to the Jacobian conjecture. This theory gives a unified frame for some results on polynomial maps that where notrelated before. A well known theorem of J. Hadamard gives a necessary and sufficient condition on a local diffeomorphism $f:R^{n} rightarrow R^{n}$ to be a global diffeomorphism.In order to show that $f$ is a global diffeomorphism it is suffices to exclude the existence of asymptotic values for $f$. The real Jacobian conjecture was shown to be false by S. Pinchuk. Our first application is to understand his construction within the general theory of asymptotic values of polynomial maps and prove that there is no such a counterexample for theJacobian conjecture over $C$. In a second application we reprove a theorem of Jeffrey Lang which gives an equivalent formulation of the Jacobian conjecture in terms of Newton polygons. This generalizes a result of Abhyankar. A third application is another equivalent formulation of the Jacobian conjecture in terms of finiteness of certain polynomial rings within $C[U,V]$. The theory has a geometrical aspect: we define and develop the theory of etale exotic surfaces. The simplest such a surface corresponds to Pinchuks construction in the real case. In fact we prove one more equivalent formulation of the Jacobian conjecture using etale exotic surfaces. We consider polynomial vector fields on etale exotic surfaces and explore their properties in relation to the Jacobian conjecture. In another application we give the structure of the real variety of the asymptotic values of a polynomial map $f:R^{2} rightarrow R^{2}$.
• Linear conditions for a polynomial $P(X,Y)$ to have younger mates: Abstract: Let $P(X,Y)=\Sigma_{0 \le i+j \le n} a_{ij}X^{i}Y^{j}$ and$Q(X,Y)=\Sigma_{0 \le i+j \le n} b_{ij}X^{i}Y^{j}$ be two polynomials. The Jacobiancondition $\partial(P,Q)/\partial(X,Y) \equiv 1$ can be interpreted as a a linearsystem of $p=\left( \begin{array}{c} 2n \\ 2 \end{array} \right)$ equations in$q=\left( \begin{array}{c} n+2 \\ 2 \end{array} \right)-1$ unknowns, the $b_{ij}$s. \Let $M_{P}$ be the matrix of the coefficients of the system. $M_{P}$ depends on $P(X,Y)$only. Let $H_{P}$ be the matrix which consists of the first $p-1$ rows of $M_{P}$ andlet $e_{p}$ be that $p \times 1$ unit vector that has a 1 in its $p$`th coordinate. \In this paper we prove that the following conditions are equivalent: \\(1) $P(X,Y)$ has a younger Jacobian mate. \(2) $rank\,\{M_{P}\} = rank\,\{M_{P}e_{p}\}$. \(3) $rank\,\{M_{P}\} = 1+rank\,\{H_{P}\}$.
Publications:
• Peretz, R.. The geometry of the asymptotics of polynomial maps. The Israel Journal of Mathematics 105: 1-59 (1998)
• Peretz, R.. Mappings With a Composite Part and With a Constant Jacobian. Applied Math. Letters 11(3): 39-43 (1998)
• Peretz, R.. On counterexamples to Keller's problem. The Illinois Journal of Mathematics 40(02): 293-303 (1997)
• Schmid, J. & Peretz, R.. The zero set of certain complex polynomials. Israel athematical conference proceedings 11: 203-208 (1997)
• Peretz, R.. Polynomial parametrization and etale exoticity. The Illinois Journal of Mathematics. 40(03): 502-517 (1997)
• Peretz, R.. The variety of the asymptotic values of a real polynomial etale map. Journal of Pure and Applied Algebra. 106(1): 103-112 (1997)
Keywords:Algebraic Geometry, Polynomial Mappings, Jacobian Conjecture, Affine Surfaces, Etale Mappings, Etale Exotic Surfaces.
Phones:
1. Phone: 972-8-6472712
2. Fax: 972-8-6472910
 Email: ronenp@math.bgu.ac.il