Some results on retracts of polynomial rings
In this paper, we first consider the relationship between a polynomial ring B over a Noetherian domain R and the ring of invariants A of a Ga-action on B, when A occurs as a retract of B. Next, we study retracts of a polynomial ring in general and address the questions of D.L. Costa raised in . Finally, we examine the behaviour of ideals and certain properties of rings under retractions.
An affine algebraic variety X is rigid if the algebra of regular functions K[X] admits no nonzero locally nilpotent derivation. We prove that a factorial trinomial hypersurface is rigid if and only if every exponent in the trinomial is at least 2.
Some special polynomial systems (antivandermond systems) and definition fields of their solutions are studied. In the case of four variables it is proved that definition field is an extension of degree 12 of real quadratic field.
Affine algebraic geometry studies algebraic subvarieties of complex affine space C^n. Nascent forms of the subject can be traced in many directions, but oneprominent series of developments is the description of the automorphism groupof the plane due to Jung and Van der Kulk (1942, 1953), together with the clas-sification of itsC-actions due to Gutwirth (1962) and its Cþ-actions due to Ebeyand Rentschler (1962, 1968). Affine algebraic geometry emerged as an independentsubject in the decade of the 1970s at the appearance of several celebrated results,including the topological characterization of the affine plane given by Ramanujam(1971), the cancelation theorem for curves given by Abhyankar, Eakin and Heinzer(1972), the Abhyankar-Moh Suzuki Theorem (1975), and the cancelation theoremfor the plane given by Fujita, Miyanishi, and Sugie (1979). In addition, theall-important Quillen-Suslin Theorem (1976), coming from a larger algebraicframework, supplied a powerful new tool for research in the subject. In this samedecade, Nagata (1972) published his famous automorphism ofC3with the con-jecture that it was not tame, and it was discovered that published solutions toKeller’s Problem,first posed in 1939, were incorrect. This problem, nowadayscalled the Jacobian Conjecture, remains open and continues to inspire a great dealof interest and new insight in thefield.Developments in the following decades continue apace. Lin and Zaidenberg(1983) characterized contractible plane curves, Danielewski (1989) gave coun-terexamples to cancelation for complex affine surfaces, and Schwarz (1989) gavecounterexamples to linearization for reductive group actions on affine space. ForCþ-actions on affine space, Roberts (1990) showed that the ring of invariants is notgenerally offinite type, and Winkelmann (1990) constructed free actions which arenot translations. Koras and Russell (1997) proved linearization ofC-actions onC3with the aid of Makar-Limanov’s work on absolute constants. Their work led tonew examples of exotic affine spaces, and the Makar-Limanov invariant is now animportant tool in classifying affine varieties. Kaliman showed that polynomial mapsonC3with generalC2-fibers are variables (2002), and that every freeCþ-action on C3is a translation (2004). Shestakov and Umirbaev (2004) showed that Nagata’sautomorphism is not tame, thus confirming Nagata’s conjecture.One can study affine algebraic geometry overfields other thanC, in particular,over an algebraically closedfieldkof positive characteristics. Russell (1980)showed that cancelation holds for the affine plane overk, while Asanuma and Gupta(1987, 2014) showed that cancelation does not hold for higher dimensional affinespaces overk. Their constructions are based on exotic line embeddings in the planedue to Segre (1956/57), and their proof makes use of the Makar-Limanov invariant.The foregoing description of significant developments is far from complete. Foran extended survey, see the article of M. Miyanishi,Recent developments in affinealgebraic geometry, in the volumeAffine Algebraic Geometry, published by OsakaUniversity Press (2007). Areas of ongoing research include cancelation andembedding problems, the Abhyankar Sathaye Conjecture,flexible varieties, auto-morphism groups and invariant theory, classification of affine varieties, lineariza-tion ofC-actions, characterization of affine spaces, and the Dolgachev-WeisfeilerConjecture.This volume presents recent advances in thefield of affine algebraic geometrywhich were featured in the talks of the conference,Polynomial Rings and AffineAlgebraic Geometry, held at Tokyo Metropolitan University in 2018. In organizingsuch a conference, we sought not only to bring together established researchers inthis area, but also to invite students, early career researchers, and those less familiarwith the area to explore and contribute to its rich content and history.
A “rational” version of the strengthened form of the Commuting Derivation Conjecture, in which the assumption of commutativity is dropped, is proved. A systematic method of constructing in any dimension greater than 3 the examples answering in the negative a question by M. El Kahoui is developed.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.