Let be a field. Let us think of the affine plane as the algebraic variety with coordinate ring and the affine line as the algebraic variety with coordinate ring . (If is algebraically closed it is safe to think and .) Then the zero-set of the polynomial is a closed subvariety of isomorphic to , and so is the zero-set of the (non-linear) polynomial . Here we obviously have an automorphism of that sends to , and it is natural to ask whether this is always the case. Along the way it is natural to inquire about the nature of the group of all automorphisms of .
The paper “Embeddings of the line in the plane” by S. S. Abhyankar and T.-T. Moh [AM1] deals with these questions. For reasons that will become clear, it is usually called the “Epimorphism Paper”. The earlier paper, “Newton-Puiseux expansion and generalized Tschirnhausen transformation” [AM2] will be termed the “Expansions Paper”. It gives a comprehensive treatment, very novel at the time, of “plane curves with one place at infinity”. It is an essential predecessor to [AM1] and has found far reaching other applications, such as the Jacobian problem, for instance. Both papers were very influential.
Many of the topics of this review were intensely discussed in Abhyankar’s circle during the Séminaire de Mathématiques Supérieures of 1970 at the Université de Montréal. The Séminaire is vividly remembered by the participants for the great intellectual ferment surrounding it, mathematical and otherwise. The lecturers were Shreeram Abhyankar, Michael Artin, Alexandre Grothendieck and Masayoshi Nagata. This was a time of great abstraction in Algebraic Geometry, but also a time of newly heightened interest in concrete and seemingly elementary problems, some very famous for being as hard to solve as easy to formulate.
Both the “Jacobian Problem” and the “Lines in the Plane Problem” belong in that class. For the lines it was accepted that the case of “one characteristic pair” (relatively prime - and -degrees) was doable, and Oscar Zariski had suggested to Abhyankar that the case of bi-degrees having a prime GCD, or a bit more generally, those with “two characteristic pairs”, be seriously considered. The brilliant full solution coming soon after, not only settled a fundamental mathematical problem, but it also led to a veritable explosion in related research activity. The result clearly fascinated a large number of mathematicians. Many additional proofs were published with techniques from topology, algebraic surface theory, complex variables, alternate arrangements bypassing or modifying the Newton-Puiseux expansions of [AM2], and so on. Some of these are mentioned below when needed as a reference, but we did not try to present a full list. Of course, all alternate proofs are valuable and may lead to new developments. The interested reader can easily locate variant proofs by a simple data base search.
Shreeram Abhyankar was our friend, guru, mentor and teacher. We dedicate this review to his inspiring mathematical career.
The Epimorphism Paper
Let be a field of characteristic and let be indeterminates over .
Main Theorem. Let and be non-constant polynomials of degree and in with coefficients in . Assume that . Assume also that either or is not divisible by . Then either divides or divides .
We will also describe the conclusion of the theorem by the phrase “ is principal”.
Using the theorem, we can deduce that if there is a so that with . Clearly . If , then we find so that . By induction on , this gives the
Epimorphism Theorem. Let be the -epimorphism with and . Let be any -epimorphism such that at least one of and is not divisible by . Then there exists an automorphism such that .
Call an automorphism elementary if or where and are non zero. Call tame, if it is a composite of elementary automorphisms.
It is then deduced that:
Addendum to the Epimorphism Theorem. can be chosen to be a tame automorphism.
The following lemma is crucial in linking the epimorphism problem to the expansion techniques of the earlier paper [AM2]. In [AM1] it is given a very elementary treatment in the spirit of High School Algebra. It will reappear later in a more sophisticated form in a general discussion of plane curves with one place at infinity.
We will call a polynomial pre-monic in of degree if with .
Lemma. Let be of -degree . Let be such that . Let be the homomorphism with and . Then a generator of is pre-monic of degree in and irreducible in . This is also described as, is a curve with one rational place at infinity.
We can, of course, choose to be monic in , but let us note that unless and hence , is also pre-monic in of some degree .
The following is a more geometric, and equivalent, version of the Epimorphism Theorem. It explains the title of the Epimorphism Paper.
Embedding Theorem. Let be a closed curve in the affine plane . Assume that is biregularly isomorphic to the affine line . Let be a generator of its ideal in and assume that either or is not divisible by . Then is a variable in , i.e., there exists such that . Moreover, the automorphism defined by is tame.
The first counterexample to the above theorems in case and the gcd of and is divisible by was, to our knowledge, given by B. Segre in [Se]. Abhyankar and Moh give a family of counterexamples of the same type. Let us describe the simplest one:
Consider and in . Then . Let be the homomorphism with and . Then is a generator of , which is monic in . Taking and prime to we have a counterexample to the Main Theorem.
Interest in the automorphism group of is older than the above results. We have the following theorem, proved by H. W. E. Jung in 1942 [J] for and by W. van der Kulk in general in 1953 [VdK]. It can now be deduced from the embedding theorem when .
Automorphism Theorem. Any automorphism of is tame.
Van der Kulk also proves a uniqueness theorem for the decomposition of a given automorphism into elementary ones. This was further elaborated by M. Nagata in [Na2].
In a letter dated 25 April 1971, at the time the news of the solution of the epimorphism problem first spread, Abhyankar was informed by G.M. Bergman that as a consequence of the commutative result the epimorphism theorem also holds in the non-commutative case. We have
Non-Commutative Epimorphism Theorem. Let and be the free associative algebras. Then the Epimorphism Theorem holds with and replacing an .
The Expansions Paper
Curves with One Place at Infinity
A place at infinity of an affine irreducible curve with coordinate ring and function field is a valuation ring of which does not contain .
We say that has one place at infinity if there is exactly one place at infinity and moreover the residue field of the valuation ring coincides with , in other words, the valuation is “residually rational”. Important examples are what Abhyankar called polynomial curves, i.e., curves parametrized by polynomials in one variable.
For a curve defined by a polynomial a place at infinity is concretely described by what Abhyankar, tongue in cheek, called Newton’s theorem on Puiseux expansions, or “meromorphic branches” of . We will call them Newton-Puiseux (NP) expansions. These are substitutions , where and (i) , (ii) is a finite algebraic extension of and (iii) is not divisible by the characteristic the gcd of and the support of is 1. This defines the valuation ring consisting of all rational functions for which .
For a plane curve defined by with, say, not divisible by the characteristic, it can be deduced that it has one rational place at infinity (sometimes shortened to just “ is a one place curve”), if and only if it has an NP expansion with and .
Note that a polynomial with one place at infinity is irreducible in and irreducible as well in the larger ring . It can be shown that is pre-monic in any choice of variables (which are not absent from ). In particular, the above equivalence is not dependent on our choice of coordinates, up to possibly switching of and . For simplicity we will assume hereafter that is monic in and .
It is a classic procedure to associate certain “characteristic sequences” to elements of . The basic tool in [AM2] is a certain clever reorganization invented by Abhyankar of such sequences associated to any meromorphic power series . Abhyankar used to relate that he recorded several ideas about plane curves in his personal notes of Zariski’s lectures on curves, but always presumed that he had simply learned them in Zariski’s course. He discovered that they were his own inventions only when Zariski was not aware of them. His idea of introducing the -sequence, for instance, made the complicated formulas involved in manipulating Newton-Puiseux expansions into much simpler statements of invariance. He wrote two separate papers entitled Inversion and Invariance of Characteristic Pairs exploring the power of these techniques.
Following [A] 6.4, we give a brief outline. Let and be given. Let be the support of and assume that the gcd of with is 1.
Start with and . Set . Define as the first element of not divisible by . Set . Continue until (since then no can be found).
This m-sequence marks the places of gcd-drops in and the d-sequence gives the successive gcd’s.
A sequence equivalent to the -sequence, but more useful, is the q-sequence defined by and thereafter.
We define the s-sequence and r-sequence by and . These play a crucial role in computing intersection numbers, as exemplified by the following kind of computation which plays an important role in the Abhyankar-Moh theory:
Consider a deformed initial part of the series , namely , agreeing with in all terms preceding and with an indeterminate. Then it is easy to calculate that the initial term of the product , where ranges over the -th roots of unity, is of the form , where is a non-zero polynomial in .
Where necessary, we will indicate the dependence on and in these definitions by a suitable notation.
Note that each of the -, - and -sequence have the associated -sequence as sequence of successive gcd’s.
One Place and Expansions
Let be a one place curve with NP expansion . We get the well-known induced factorization , where is a primitive -th root of unity. In particular, all roots of have the same support and the characteristic sequences we defined depend on only. We note that .
We mention two essential ingredients of the Abhyankar-Moh theory which are responsible for most of its successes.
The Irreducibility Criterion of Abhyankar and Moh. Assume that and is monic in . Then has one place at infinity iff there is a “test series” such that
Moreover, given any series passing this test, there is a “root” satisfying and .
This Lemma, originally in [AM2], was later reproved by Abhyankar in greater detail in [A2].
The Innovation of the Approximate Roots. Let have one place at infinity, arranged to be monic in . Consider the characteristic sequences as described above for an NP expansion . Then for each , , we get approximate roots defined by
For , and for , is monic in of degree and
Such polynomials are easily seen to be uniquely defined by for any factors of , but for the chosen from the characteristic sequence, it is shown that each is a curve with one place at infinity and that , taken mod , has value in the valuation at infinity of .
This implies that an NP expansion of matches that of up to . The idea that this should happen for was first put forward by Moh and perfected to the above form by the genius of Abhyankar.
The following result is now recognized as describing a fundamental property of one place curves.
The One Place Theorem for Translates of a One Place Curve. If and has one place at infinity, then also has one place at infinity for any . Moreover, all translates have NP expansions that match through the last characteristic term. In geometric language, this means that and go through each other at infinity at all the singular points in a sequence of quadratic transforms.
This is deduced from the irreducibility lemma and the explicit calculation of the initial forms in terms of the approximate roots. Over it implies that the fibration defined by is topologically trivial in a neighborhood of infinity.
The Value Semigroup of a One Place Curve, Standard Basis
Let and have one place at infinity with NP expansion and coordinate ring . Let be the canonical homomorphism with . The valuation at infinity on the quotient field of is defined by . We define the value semigroup of as . Define and put . For let , where the are the approximate roots introduced above and, as we said, . Let be the set of -tuples of integers with and for . The following results are the technical core of the Abhyankar-Moh theory:
. These “restricted expansions” are unique.
The “standard monomials” provide a -basis of . Note that in each linear combination over of standard monomials there is a unique term whose valuation at infinity is the valuation of the sum.
It can be deduced from the above that if belongs to , then is principal and hence one of divides the other. We obtain a proof of the Epimorphism Theorem since in that case is .
We remark that expansion techniques can also be applied to irreducible elements of the power series ring , in particular when it is the completion of of the local ring at the point at infinity in of a one-place curve. If one does not insist that the expansion be of NP type (with one of the variables a power of the parameter ), the basic definitions can be made without explicit reference to the characteristic . This is useful in some applications, see [Ru3].
Suzuki’s Proof A proof of the Embedding Theorem contemporary with that of Abhyankar and Moh was given by M. Suzuki [Su]. It is very different in spirit, and the Embedding Theorem nowadays is often cited as the AMS-Theorem. Suzuki’s paper also was very influential. It uses methods of complex analysis, in particular the theory of pluri-subharmonic functions, to study polynomial maps , where is an irreducible polynomial. A key result, now usually referred to as Suzuki’s formula, is that the topological Euler characteristic of any singular (special) fiber is at least as big as that of a regular (general) fiber. (This generalizes a fact well known in the case of proper maps.) In case , Suzuki then goes on to show that is in fact a regular fiber. His methods apply to morphisms for surfaces more general than , in particular all smooth affine surfaces. His results have been extended and sharpened by M. Zaidenberg [Za], and a proof of Suzuki’s formula relying on geometric methods rather than complex analysis, or, let us say, more accessible to algebraic geometers, has been given by R. Gurjar [Gu2].
Positive Characteristic The counterexamples to the Embedding Theorem in positive characteristic cited above left open the question whether at least the “One Place Theorem” (see 3.2) still holds. This question was raised by Abhyankar in [A]. A partially positive, partially negative answer was given by R. Ganong [Ga1]. For simplicity we assume that is algebraically closed.
The Generic One Place Theorem. Let have one place at infinity. Then the generic member , transcendental over , of the pencil , has one place at infinity with residue field purely inseparable over . Moreover, for almost all , has one place at infinity with multiplicity sequence at infinity the same as that of over the algebraic closure of .
This is a “best possible” result: Ganong also gives examples of one place curves with with a special member having more than one place at infinity, or with general member having a multiplicity sequence at infinity different from that of . Following the lead of M. Nagata [Na1] and M. Miyanishi [Mi1], Ganong investigates special properties of the pencil obtained by eliminating the base points at infinity of the pencil . Here the fiber at infinity is simply connected (it is a tree of curves isomorphic to the projective line), and a key result is a positive characteristic version of a lemma of Kodaira [Kod] on the global multiplicity of such fibers. We remark that in case the place at infinity of is rational over (i.e., the residue field is , e.g., if ), then all have the same multiplicity sequence and the same infinitely near multiple points at infinity. In particular, the Embedding Theorem follows if is a line.
Lines in the plane in positive characteristic are still poorly understood. Some contributions to the question have been made by [Da2, Da3, Ga1, Mo2]. The following closely related conjectures seem to have been made by several researchers. An overview of relevant results is given in [Ga3], see also [Mi2].
Lines Conjectures in Characteristic .
If is a line, then all are lines.
If is a line, then the relative Frobenius w.r.t. is a plane, i.e., is a polynomial ring.
Further Developments from the Expansions Paper The Jacobian Problem Inspired by their new machinery, Abhyankar and Moh produced independently several papers attacking the famous Jacobian Problem. In dimension two, it asks if polynomials in the polynomial ring over a field of characteristic zero with Jacobian determinant 1 generate .
Indeed, this problem was rejuvenated and popularized by Abhyankar along with several other problems in Affine Geometry of two and three dimensions as a way to attract new students to important but accessible problems in Algebraic Geometry.
Abhyankar and Moh propose to consider as elements of , that is as defining a polynomial curve over with serving as parameter, see [A] for details. They quickly translated the Jacobian condition into conditions on the resulting NP expansion and produced the following striking result, among others:
The Two Point Theorem. The Jacobian condition implies that have at most two points at infinity, i.e., their top degree forms in have at most two non-associate factors. Moreover, if it can be deduced that the Jacobian condition implies that have at most one point at infinity, then the Jacobian problem has an affirmative answer.
For brevity, we stop here, but the problem has a long colorful history and many results (even in higher dimensions) are available at the touch of a key stroke!
Finiteness of Embeddings of One Place Curves One way of stating the epimorphism theorem is to say that there is only one equivalence class of embeddings of an affine line in the affine plane up to automorphisms of the plane. Abhyankar raised the corresponding question for general one place curves: Suppose that are two epimorphisms from onto the coordinate ring of a plane curve with one rational place at infinity. Does it follow that and are equivalent? If not, is it at least true that the number of equivalence classes is finite?
Let be a generator of , say. We can then arrange by an automorphism of that is non-principal. This gives that is a number not in the value semigroup, by the non principal condition (see 3.3). In [ASi] Abhyankar and Singh prove the following striking result: Two embeddings are equivalent if and only if the corresponding ’s are equal. This, combined with the fact that there are only finitely many negative numbers not in the value-semigroup, gives the finiteness of embeddings with a very explicit bound on the number. (We have tacitly assumed . Otherwise even the line has infinitely many inequivalent embeddings [Ga1].)
Planar Semigroups The properties of the characteristic -sequence in the Abhyankar-Moh theory can be codified abstractly and semigroups generated by an -sequence as in 3.3 have been called planar semigroups by Sathaye. (Sathaye actually preferred to work with the negative of an -sequence.) It was announced in [Sa2] and shown in [SS] that every planar semi-group is the value semi-group of a one place curve. See [A2] as well. The irreducibility criterion 3.2.1 plays a significant role here.
An important question, originally raised by Abhyankar himself, is to characterize the semi-groups at infinity, or equivalently the degree semigroups, of plane polynomial curves. This remains unsolved to date. For further calculations and conjectures about these, see [SS], [SFY] and [M-L]. An interesting special case is the Lin-Zaidenberg Theorem [LZ] which asserts that there is only one class for polynomial curves with only unibranch singularities. (The theorem proves more, namely that such curves have only one quasi-homogeneous singularity and that .)
A Sampling of Further Related Results The Epimorphism Theorem inspired a large amount of research on related questions. We can only give a brief sampling. Coefficient rings more general than fields were considered [Ba], [RS], [Ve]. More general closed embeddings of affine -space in affine -space were studied [Kal1], [Sr]. Abhyankar in particular advocated the study of embeddings of lines and planes in affine 3-space. For some results on lines see [A3], [AS], [Cr], [BR], [Sh], and for planes see [Sa1], [Ru2], [Wr], [San]. These papers depend in a crucial way on the Epimorphism Theorem. An unexpected generalization of the Epimorphism Theorem was developed by Sathaye in [Sa3], [Sa4] and [Sa5]. It became an important tool in several studies of -fibrations over curves. The Epimorphism Theorem also is the essential ingredient in the proof of a special case of the linearization conjecture for -actions on , see [KKM-LR].
It was a suggestion coming out of Abhyankar’s Purdue seminar to consider closed curves in the affine plane with several places at infinity. A once punctured affine line, when , is the obvious first candidate, see [BZ], [C-NKR], [Kal2], [Ko2] for results in this case. Another suggestion was to investigate “field generators” (instead “ring generators” as in the Epimorphism Theorem), that is, polynomials that together with a complementary rational function generate the field , see [Ja], [NN], [MS], [Ru1], [Da1].
The Epimorphism Theorem gave a strong boost to Affine Algebraic Geometry, the study of algebraic varieties closely related to affine spaces, in particular affine rational surfaces. The study of affine lines on such surfaces became an important part of their classification via logarithmic Kodaira dimension [It], [GMMR], [GM1], [KK]. In turn, the classification theory has been used to prove the Epimorphism Theorem [GM2], [Gu1], [Ko1].
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