The Cambridge Tracts are “devoted to thorough, yet reasonably concise, treatments of topics in any branch of mathematics.” And so Analysis in Positive Characteristic, by Anatoly N. Kochubei of the NAS of Ukraine, comes in at 210 pages, and covers the following themes: orthornormal systems, calculus, differential equations, special functions, and Carlitz rings.
Judging by the first four of these headings we might be on familiar ground, material every mathematician learns pretty much in the crib (well, maybe not for special functions like the hypergeometric ones… but still, we are not dealing which such alien things as, say, topoi or Lie algebroids). Only Carlitz rings appear exotic, and upon looking up what they are (on p. 173 ff. of the book under review) we find out that indeed they are such. Suffice it to say for now that “[i]n many respects [they] may be seen as the function field counterparts of the Weyl algebras of polynomial differential operators.” With this characterization we encounter the crux of the matter: we are dealing with function fields, manifestly over fields of positive characteristic, and so subjects like calculus and differential equations take on an entirely different appearance (with (formal) Laurent series taking central stage).
Accordingly the setting for the counterpart to analysis over R or C proffered here is that of the ring of power series with finite singular parts over Fq, where q is a power of a prime, and it turns out that orthonormality drives the car. Says Kochubei (p.1): “[we] study [the] two most important orthonormal systems appearing in this theory, the Carlitz polynomials and hyperdifferentiations. Their role, especially that of the Carlitz polynomials, will be crucial throughout this book.”
Kochubei’s treatment is very thorough and his presentation of the material chosen is both readable and accessible. It can’t be helped of course that things get denser as we go along in the book: the Carlitz rings are not easy to play with. On the other hand, the stage is certainly set for them.
Kochubei himself is a serious player in the game, and his book shows this. It is solid scholarship well worth the price of admission.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.