defines, if we interpret “√x” as two-valued, a function with four branches
±√x±√9x={ −4√x,−2√x,2√x,4√x}.
It is a rigorous mathematical fact [302] that solutions of equa-
Chris Hobbs aged 6 tions of degree higher than two cannot be analytically expressed
by choiceless multivalued formulae (even if we allow for more so- phisticated analytic functions than radicals); see a discussion of the topological nature of this fact by Vladimir Arnold [3, p. 38].
This last observation is especially interesting in the historic context. At the early period of development of symbolic algebra, mathematicians were tempted to introduce functions more general than roots. The following extract from Pierpaolo Muscharello’sAl- gorismusfrom 1478 is taken from Jens Høyrup [52]:
Pronic root is as you say,9times9makes81. And now take the root of9, which is3, and this3is added above81, so that the pronic root of84is said to ne3.
In effect, Muscharello wanted to introduce the inverse of the func- tion
z7→z4+z.
Arnold’s theorem explains why such tricks could not lead to an easy solution of cubic and quadric equations and had been abandoned.
1.5 You name it—we have it
This section is more technical and can be skipped.
As I have already said on several occasions, this book is about simple atomic objects and processes of mathematics. However, mathematics is huge and immensely rich; even the simplest ob- servations about its simplest objects may already have been de- veloped into sophisticated and highly specialized theories. Mathe- matics’ astonishing cornucopian richness and its bizarre diversity are not frequently mentioned in works on philosophy and method- ology of mathematics—but this point has to be emphasized, since its makes the question aboutunityof mathematics much more in- teresting.
In this section, I will briefly describe a “mini-mathematics”, a mathematical theory concerned with a close relative of the abso- lute value function, themaximumfunction of two variables
z= max(x, y).
Of course, the absolute value function|x|can be expressed as
|x|= max(x,−x).
Similarly, the maximum max(x, y) can be expressed in terms of the absolute value|x|and arithmetic operations—I leave it to the
The theory is known under the name oftropical mathemat- ics. The strange name has no deep meaning: the adjective “trop- ical” was coined by French mathematicians in the honor of their Brazilian colleague Imre Simon, one of the pioneers of the new discipline. Tropical mathematics works with usual real numbers but uses only two operations: addition, x+y, and taking the maximum,max(x, y)—therefore it is one of the extreme cases of “switch-flipping”, choice-based mathematics. Notice that addition is distributive with respect to taking maximum:
a+ max(b, c) = max(a+b, a+c).
This crucial observation is emphasized by renaming the two basic operations into new “multiplication” and “addition”:
a¯b=a+b, a⊕b= max(a, b).
The previous identity takes the more familiar shape of the dis- tributive law:
a¯(b⊕c) =a¯b⊕a¯c.
Of course, operations ¯and ⊕are commutative and associative. After recycling the traditional shorthand
xn=x¯ · · · ¯x (ntimes),
we can introduce polynomials as well as matrix multiplication, determinants, etc. For example, the tropical determinant of the matrixA = (aij)is defined by evaluating the expansion formula (see page 8) tropically and ignoring the signs of permutations σ involved in the classical formula for determinant [392, 404]:8
dettrA= M σ∈Symn (a1,σ(1)¯ · · · ¯an,σ(n)) = max σ∈Symn(a1,σ(1)+· · ·+an,σ(n))
Here the “sum” is taken over the setSymnof all permutations of indices1, . . . , n; for example,
dettr ¯ ¯ ¯ ¯a bc d ¯ ¯ ¯ ¯= max(a+d, b+c).
Therefore the “sum” involves n!“monomials”—exactly as in the case of ordinary determinants. Let us return for a second to our discussion of the complexity of evaluation of determinants in choice- based and choiceless models of computation. It is not obvious at all that a tropical determinant can be evaluated usingO(n3)ele-
mentary operations—but it is true. The evaluation of the tropical determinant is the classicalassignment problem in discrete opti- mization and the bounds for its complexity are well-known [396, Corollary 17.4b]. Indeed, finding the value of the tropical determi- nant amounts to finding a permutationi7→σ(i)which maximizes the sum
a1,σ(1)+· · ·+an,σ(n).
This is an old problem of applied discrete optimization: a company has boughtnmachinesM1, . . . , Mnwhich have to be assigned to
1.5 You name it—we have it 13
nfactoriesF1, . . . , Fn; the expected profit from assigning machine Mito factoryFjisaij; find the assignment
Mi7→Fσ(i)
which maximizes the expected profit a1,σ(1)+· · ·+an,σ(n).
Chris Hobbs aged 16
Fig. 1.1.A cubic curve in the tropical projective plane (adapted from Mikhalkin [381]).
We also have a full-blown tropical algebraic geometry, where curves in the plane are made from pieces of straight lines (Fig- ure 1.1)—quite like the graph of the absolute value functiony =
|x|, the starting point of our discussion.
Tropical mathematics is amusing, but is it relevant? Yes, and very much so. Moreover, it currently is experiencing an explo- sive growth. There are intrinsic mathematical reasons for tropical mathematics to exist, but its present flourishing is largely moti- vated by applications.
One application is mathematical genomics: tropical geometry captures the essential properties of “distance” between species in the phylogenic tree.
Another is theoretical physics: tropical mathematics can be treated as a result of the so-called Maslov dequantization of tradi- tional mathematics over numerical fields as the Planck constant~ tends to zero taking imaginary values [375].
A third one is computer science and the theory of time-dependent systems, like queuing networks (where tropical mathematics is known under the name of a(max,+)-algebra). The rationale be- hind this class of applications is an observation so simple and banal that it has a certain camp value. Indeed, we do not nor- mally multiply time by time; instead, we either add two intervals of time (which corresponds to consecutive execution of two pro- cesses) or compare the lengths of two intervals—to decide which
process ends earlier. Therefore tropical mathematics is mathemat- ics oftime—which also explains its applications to genomics: phy- logenic trees grow in time, and the geometry of phylogenic trees reflects the geometry of time.