Nonlinear algebra

What is nonlinear algebra?

Negations are tricky. They may be the largest source of bugs in database queries. You have to carefully think about what exactly are you negating.

Any time you see “non-” attached to something, you have to ask what the context is in which the negation takes place. For example, if you hear someone say “non polar bear,” do they mean a non-polar bear, i.e. any bear besides a polar bear, such as a grizzly bear? Or do they mean non-polarbear, i.e. anything that isn’t a polarbear, such as a giraffe or an espresso machine?

Another subtlety is whether “non” means “definitely not” or “not necessarily.” I specialized in nonlinear PDEs in grad school, and “nonlinear” always meant “not necessarily linear.” I don’t recall ever seeing a theorem that required an operator to not be linear, only theorems that did not require operators to necessarily be linear.

With that introduction, what could nonlinear algebra mean? To many people, it would mean working with equations that contain nonlinear terms, such as solving sin(x) = 2x. The sine function is certainly nonlinear, but it’s not the kind of function algebraists are usually interested in.

To a large extent, the algebraist’s universe consists of polynomials. In that context, “nonlinear” means “not necessarily linear polynomials.” So things that are nonlinear in the sense that x³ is nonlinear, not in the sense that sin(x) is nonlinear.

If you think of linear algebra as the mathematics surrounding the solution of systems of linear equations, nonlinear algebra is the math surrounding the solution of systems of polynomial equations. Note that this uses “non” in the “not necessarily” sense: polynomials that are not necessarily linear. Linearity is not assumed, but it’s not excluded either.

You might be thinking “isn’t that just algebraic geometry?” if you’re familiar with algebraic geometry. Strictly speaking the answer is yes, but there’s a difference in connotation if not denotation.

Nonlinear algebra, at least how the term has been used recently, means the application-motivated study of ways to solve polynomial equations. As a recent book review puts it,

Nonlinear algebra’s focus is on computation and applications, and the theoretical results that need to be developed accordingly. Michalek and Sturmfels explain that this name is not just a rebranding of algebraic geometry but that it is intended to capture this focus, and to be more friendly to applied mathematicians, questioning the historic boundaries between pure and applied mathematics.

I like the term nonlinear algebra. Algebraic geometry is a vast subject, and there needs to be a term that distinguishes solving polynomial equations for applications from studying some of the most esoteric math out there. One could argue that the latter is ultimately the proper theory of the former, but a lot of people don’t have the time or abstraction tolerance for that.

The term nonlinear algebra may be better than applied algebraic geometry because the latter could imply that the goal is to master the entire edifice of algebraic geometry and look for applications. Nonlinear algebra implies it is for someone looking to learn math as immediately useful for solving nonlinear equations as linear algebra is for solving linear equations.

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