I am wondering if it would be worthwhile to ask why a certain textbook and/or its authors (Dolotin and Morozov) are not well known.

From the description of the book:

This unique text presents the new domain of consistent non-linear counterparts for all basic objects and tools of linear algebra, and develops an adequate calculus for solving non-linear algebraic and differential equations.

Is it pseudo-mathematics?

This article by the authors is a concise introduction: http://arxiv.org/abs/hep-th/0609022

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    $\begingroup$ Why do you want to know why the book is not well known? Does the book contain some ideas that are not common elsewhere, or something? There are many books that are not well known, not for any particular reason, so I fail to see your motivation. $\endgroup$ – Joonas Ilmavirta Mar 11 '15 at 17:28
  • $\begingroup$ @JoonasIlmavirta From the description of the book: "This unique text presents the new domain of consistent non-linear counterparts for all basic objects and tools of linear algebra, and develops an adequate calculus for solving non-linear algebraic and differential equations." -- surely, something like this, an algebra for non-linear equations that parallels the power of linear algebra would be very curious to all? So, that's why I wonder if its just pseudo-mathematics, and the claim of the book is simply false. $\endgroup$ – user89 Mar 11 '15 at 17:49
  • $\begingroup$ Responding to the previous comment: outlandish claims don't always attract interest from people who have enough on their plates to deal with $\endgroup$ – Yemon Choi Mar 11 '15 at 17:50
  • $\begingroup$ @YemonChoi Is it safe for me to ignore this text too then? I guess that's what I want to understand. $\endgroup$ – user89 Mar 11 '15 at 17:51
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    $\begingroup$ If the book claims to give tools for nonlinear equations that are just as strong as the usual ones for linear equations, it is certainly too good to be true. It may contain some clever methods, but the description claims too much, which makes me lose interest. $\endgroup$ – Joonas Ilmavirta Mar 11 '15 at 17:55
  • $\begingroup$ @JoonasIlmavirta Thanks! That's exactly what I needed to understand. $\endgroup$ – user89 Mar 11 '15 at 17:58
  • $\begingroup$ I wrote my thoughts into an answer to address similar questions in the future. $\endgroup$ – Joonas Ilmavirta Mar 11 '15 at 18:14
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    $\begingroup$ The Zentralblatt review quoted on the Amazon page says "Reading the text needs motivation, experience and endurance,..." and I think this goes a long way to making a book obscure, whatever its merits. (I checked the full review, to see if it was in any way negative, and it wasn't, but the review isn't full of praise, either.) $\endgroup$ – David Roberts Jan 24 at 22:26

Questions like why is something or someone not well known are not well posed mathematical questions, and are therefore likely to be closed as primarily opinion-based. If you can modify such a question to something like "Why is this appealing method not useful?", it might be more welcome; just make sure that the question is actually a question, not an start of a discussion and it's probably fine. These pages at our help pages give a good idea of what kinds of questions work here.

There are many textbooks out there, and not all of them are well known. Therefore I would find it more reasonable to ask why some book is well known, but such questions also have the restrictions mentioned above.

I don't know the book you were planning to ask about, but it seems to claim too much. The description makes the book sound like a full treatment of nonlinear algebra, which is certainly not possible. It may be useful to some and contain clever ideas, but the description you cite claims too much. A book need not be wonderful even if its cover claims so – reliable evidence for greatness can only be something external to the book. (And this applies to all books, not just mathematical ones.)


I cannot judge right now the merits or demerits of the specific book you mention. Joonas Ilmavirta addresses in their answer whether your question is appropriate and contains some remarks about the book's claims. The term nonlinear algebra has been used by others though. See, for instance, the draft text by Michalek & Sturmfels, the research group at MPI Leipzig, and Sturmfels' lecture series.

Linear algebra is the foundation of much of mathematics, particularly in applied mathematics. Numerical linear algebra is the basis of scientific computing, and its importance for the sciences and engineering can hardly be overestimated. The ubiquity of linear algebra masks the fairly recent growth of nonlinear models across the mathematical sciences. There has been a proliferation of methods based on systems of multivariate polynomial equations and inequalities. This is fueled by recent theoretical advances, efficient software, and an increased awareness of these tools. At the heart of this lies algebraic geometry, but there are links to many other branches, such as combinatorics, algebraic topology, commutative algebra, convex and discrete geometry, tensors and multilinear algebra, number theory, representation theory, and symbolic and numerical computation. Application areas include optimization, statistics, complexity theory, among many others.

Nonlinear algebra is not simply a rebranding of algebraic geometry. It is a recognition that a focus on computation and applications, and the theoretical needs that this requires, results in a body of inquiry that is complementary to the existing curriculum. The term nonlinear algebra is intended to capture these trends, and to be more friendly to applied scientists. A special research semester with that title, held in the fall of 2018 at the Institute for Computational and Experimental Research in Mathematics (ICERM) in Providence, Rhode Island, explored the theoretical and computational challenges that have arisen, and it charted the course for the future. This book supports this effort by offering a warm welcome to nonlinear algebra.

(From pp. xi-xii of the draft of Invitation to Nonlinear Algebra by Michalek & Sturmfels)


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