Shreeram S. Abhyankar

On the uniqueness of the coefficient ring in a polynomial ring

If k is a field and X and Y are indeterminates then the statement “consider R = Iz[X, Y] as a polynomial ring in one variable” is ambiguous, for there arc infinitely many possible choices for the ring of coefficients (e.g., If A, = k[X f Y”] then A,[Y] == B,,[Y] 7: R but A,, f J,, if m # n). On the other hand, if Z denotes the integers then the polynomial ring Z[X] has a unique subring over which it is a polynomial ring. This investigation began with our consideration of the first of these examples. In fact, Coleman had asked: If k is a field, then although k[X, 1’1 can be written as a polynomial ring in many different ways, is it true that all of the possible coefficient rings are isomorphic? That is, if T is transcendental over d and 4[T] == k[X, Y], is A a polynomial ring over k ? We found that this is indeed the case (see our (2.8)).We next proved the following: If il is a one dimensional afine domain over a

Automatic parameterization of rational curves and surfaces III: algebraic plane curves

eZd and B is a ring such that A [-Xl = B[ E;] zs an equality of polynomial rings, then either A ~ B or there is afield k such that each of A and B is a polynomial +zg in one variabZe over k. This is a corollary of (3.3) in the present paper. Our (7.7) sketches a version of the original proof. In studying this argument, we found that there were implicit in it techniques for investigating the following general question: Suppose A and B are commutative rings with identity and the polynomial rings 4 [X, , . . . , X,,] and B[E; ,..., I’,] are isomorphic, how are A and B related? Are A and B isomorphic? In particular, when does the given isomorphism take i4 onto B ? This study is mainly centered on the latter portion of the question. We are concerned almost entirely with domains It is convenient to use the following terminology which is modeled after that

Galois theory on the line in nonzero characteristic

Abstract We present algorithms to compute the genus and rational parametric equations, for implicitly defined irreducible plane algebraic curves of arbitrary degree. Rational parameterizations exist for all irreducible algebraic curves of genus 0. The genus is computed by a complete analysis of the singularities of plane algebraic curves, using affine quadratic transformations. The rational parameterization techniques, essentially, reduce to solving symbolically systems of homogeneous linear equations and the computation of polynomial resultants.

Irreducibility criterion for germs of analytic functions of two complex variables

The author surveys Galois theory of function fields with non-zero caracteristic and its relation to the structure of finite permutation groups and matrix groups.

Automatic parameterization of rational curves and surfaces IV: algebraic space curves

Let S(X, Y) be the germ of an analytic function of two complex variables near the origin. To decide whether f is irreducible or not, we can apply a succession of blowing-ups to desingularize f and see if the proper transform off ever bifurcates, and then use the fact that f is irreducible iff such a bifurcation never takes place. Alternatively, we can use Newton’s theorem (also called Puiseux’s theorem) to completely factor S into linear factors in Y by allowing fractional power series in X and then collate these factors into conjugacy classes, and now use the fact that fis irreducible iff there is only one conjugacy class. In a recent conversation, T. C. Kuo of the University of Sydney in Australia asked me the following question:

NICE EQUATIONS FOR NICE GROUPS

For an irreducible algebraic space curve <italic>C</italic> that is implicitly defined as the intersection of two algebraic surfaces, <italic>f</italic> (<italic>x</italic>, <italic>y</italic>, <italic>z</italic>) = 0 and <italic>g</italic> (<italic>x</italic>, <italic>y</italic>, <italic>z</italic>) = 0, there always exists a birational correspondence between the points of <italic>C</italic> and the points of an irreducible plane curve <italic>P</italic>, whose genus is the same as that of <italic>C</italic>. Thus <italic>C</italic> is rational if the genus of <italic>P</italic> is zero. Given an irreducible space curve <italic>C</italic> = (<italic>f</italic> ∩ <italic>g</italic>), with <italic>f</italic> and <italic>g</italic> not tangent along <italic>C</italic>, we present a method of obtaining a projected irreducible plane curve <italic>P</italic> together with birational maps between the points of <italic>P</italic> and <italic>C</italic>. Together with [4], this method yields an algorithm to compute the genus of <italic>C</italic>, and if the genus is zero, the rational parametric equations for <italic>C</italic>. As a biproduct, this method also yields the implicit and parametric equations of a rational surface <italic>S</italic> containing the space curve <italic>C</italic>. The birational mappings of implicitly defined space curves find numerous applications in geometric modeling and computer graphics since they provide an efficient way of manipulating curves in space by processing curves in the plane. Additionally, having rational surfaces containing <italic>C</italic> yields a simple way of generating related families of rational space curves.

Resolution of singularities and modular Galois theory

Nice trinomial equations are given for unramified coverings of the affine line in nonzero characteristicp with PSL(m,q) and SL(m,q) as Galois groups. Likewise, nice trinomial equations are given for unramified coverings of the (once) punctured affine line in nonzero characteristicp with PGL(m,q) and GL(m,q) as Galois groups. Herem>1 is any integer andq>1 is any power ofp.

Automatic parameterization of rational curves and surfaces 1: conics and conicoids

I shall sketch a brief history of the desingularization problem from Riemann thru Zariski to Hironaka, including the part I played in it and the work on Galois theory which this led me to, and how that caused me to search out many group theory gurus. I shall also formulate several conjectures and suggest numerous thesis problems.

Computations with Algebraic Curves

Given the implicit equation for degree two curves (conics) and degree two surfaces (conicoids), algorithms are described here, which obtain their corresponding rational parametric equations (a polynomial divided by another). These rational parameterizations are considered over the fields of rationals, reals and complex numbers. In doing so, solutions are given to important subproblems of finding rational and real points on the given conic curve or conicoid surface. Further polynomial parameterizations are obtained whenever they exist for the conics or conicoids. These algorithms have been implemented on a VAX-780 using VAXIMA.

On Hilbertian ideals

We present a variety of computational techniques dealing with algebraic curves both in the plane and in space. Our main results are polynomial time algorithms (1) to compute the genus of plane algebraic curves, (2) to compute the rational parametric equations for implicitly defined rational plane algebraic curves of arbitrary degree, (3) to compute birational mappings between points on irreducible space curves and points on projected plane curves and thereby to compute the genus and rational parametric equations for implicitly defined rational space curves of arbitrary degree, (4) to check for the faithfulness (one to one) of parameterizations.