Finding Gemstones: on the quest for mathematical beauty and truth
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Tropical polynomials and your federal tax return

It’s Tax Day here in the United States, and I spent a larger portion of the past weekend than I would have liked filling out the appropriate forms. But even among tax forms and legal documents you can find a mathematical gemstone or two!

The instructions for filling out, say, IRS form 1040, are rather peculiar in that they try give the reader very simple mathematical operations to carry out, one at a time, to end up with a complicated function of many variables. As far as I could tell, the valid operations are:

  • Entering a value for a variable (e.g. income, standard deduction, etc.)
  • $+$: Addition of two entries
  • $\times$: Multiplication
  • $\min$: Taking the minimum of two entries
  • $\overset{\cdot}{-}$: A modified version of subtraction, where $a \overset{\cdot}{-} b$ is defined to be $a-b$ if $a\ge b$ and $0$ otherwise.

I found it interesting that every complicated tax computing formula can be broken down into steps of this form. In fact, it says something about the tax formulae: they can all be written as a composition of addition, multiplication, scalar multiplication, min, and $\overset{\cdot}{-}$.

The most interesting of the operations is $\overset{\cdot}{-}$, which is pronounced ``monus’’. According to the Wikipedia article on monus, the monus operation can in fact be defined on any commutative monoid $C$ in which the relation $\le$ (defined by $a\le b$ if and only if there exists $c\in C$ for which $a+c=b$) is a partial order. This is certainly true for the nonnegative real numbers, which seems to be the domain of operation of the IRS.

The nicest fact about monus in this context, however, is that we can express $\min$ in terms of it:

\[\min(a,b)=a\overset{\cdot}{-} (a\overset{\cdot}{-} b)\]

This means that every tax formula can be written as a polynomial in several variables in which we replace minus by monus! Let’s call such polynomials ``Monus polynomials”. So one example of a monus polynomial would be:

\[2xy+x^2yz\overset{\cdot}{-} 3z^3\]

This opens up a plethora of interesting questions. What are the properties of monus polynomials? Are there nice analogues of theorems such as the fundamental theorem of algebra from the classical case? Can we define monus-algebraic field extensions?

This may indeed be an interesting field of study, because tropical polynomials are a special case of monus polynomials, written using only $+$ and $\min$. Who knew that tropical geometry would arise so naturally in the study of IRS Form 1040?