In the wide area of the evaluation of a set of bilinear forms, which includes computing the products of two complex numbers, two integers, two matrices, two polynomials, and two polynomials modulo a third one (for more examples see [1]), deGroote focuses on two separate topics: a proof and extensions of Alder and Strassen’s lower bound on the multiplicative complexity M of an associative algebra (chaps. 4 and 5) and the upper bounds on the cost of multiplication of matrices of extremely large dimensions (chap. 3).

The multiplicative complexity M of a set of bilinear forms is the minimum number of nonlinear arithmetic operations in straight line schemes for computing such a set. Of course, the actual cost also depends on linear operations (additions, multiplications by constants) and on the numerical magnitudes of the operands (see [1], chap. 4), but estimating M mostly involves interesting algebraic techniques. Multiplication in associative algebras includes all the bilinear computations listed above (and many others) as special cases.

Alder and Strassen derived the lower bounds on M previously known in these cases and indicated that their proofs, although “greatly different from each other, all used the so-called basic substitution method of Pan 1966 as a basic ingredient” [2]. Then they presented a nontrivial unification of these proofs, establishing that M≥ dim (A) (the number of maximal two-sided ideals of A) in the case of any associative algebra A. DeGroote proves that remarkable theorem in Chapter 4 (after preparations in chaps. 1 and 2) and then considers (1) some cases where that bound is sharp, and (2) some algorithms reaching it. (The reader should take his term optimal for such algorithms with a grain of salt.)

The statements of the results throughout the book assume familiarity with texts on associative algebras and multilinear algebra. The clarity of the presentation could be much improved. The demonstration by examples and by specification of formal definitions is insufficient and is sometimes postponed too much (the demonstrations on pp. 26–29 and on p. 41 would be more helpful if they had been placed earlier). Some concepts are used without or before being defined (simple and semisimple algebras, p. 37, and separable algebraic field extension, p. 53), some are defined under one name and used under another (Strassen’s direct sum conjecture, p. 51, is the term from [3] (the book not cited by deGroote) for what deGroote calls rank conjecture, p. 39). There are some successful exceptions (sect. 2.2), and the book should be credited as the first expository presentation of Alder and Strassen’s work and of its extensions (including the author’s own research contributions). Unfortunately I can say nothing similiarly positive about Chapter 3 on multiplication of large matrices and section 2.6 on border rank, whose material was extensively covered earlier in other books [3,4,5] and in two survey articles [6,7] (none of the five are cited by deGroote).

Finally, the author is unfair historically. On page 25 he (naively?) attributes to Strassen 1973 (ref. 51 of deGroote) the concept of tensor rank due to Hitchcock 1927 and cited, say in [8]. His remarks on page 81 arbitrarily shorten and thus distort the history of fast matrix multiplication. Throughout the book he credits Schönhage 1981 for several important contributions of other authors, whose earlier works on border rank and trilinear aggregating were directly continued and formalized in Schönhage 1981 [9]. The latter distortion is particularly grave because both concepts of border rank and trilinear aggregating were the basis not only for Schönhage’s main theorem but also for the design of all the asymptotically fastest matrix multiplication algorithms since 1978 (including the latest ones of 1986–87).