Trevor Gunn, PhD (they/them)

Papers

Descartes's Rule of Signs

The Arithmetic of Signs

Through primary and secondary education, one learns how to add, subtract, multiply and divide rational and decimal numbers. Instead, suppose we forget everything about our numbers except whether they are positive, negative or zero and try to do similar arithmetic there.

So now are arithmetic consists of rules like: positive + positive = positive or negative ÷ negative = positive. This arithmetic is pretty straightforward to use except for the rule which says that a positive plus a negative number may be any sort of number.

Arithmetics like these are called hyperfields. In a hyperfield, we can add, subtract, multiply and divide, except addition and subtraction may produce a set of posibilities rather than a unique outcome.

Polynomials over Hyperfields

My PhD advisor, together with Oliver Lorscheid, showed that Descartes's Rule is a natural consequence of considering polynomials in the arithmetic of signs. I.e. instead of a polynomial with integer or rational coefficients, the coefficients are either "positive," "negative" or "zero." Instead of asking "is 2 a root? and with what multiplicity?" we ask "is 'positive' a root? and with what multiplicity?"

My work extends their work in two ways. In one paper, I extend their work to a broader class of arithmetics. In a second paper, with Andreas Gross, we extend their ideas to polynomials in more than one variable.

Tropical Geometry and Valued Fields

A sign is not the only information that a rational number has, it also has an absolute value. More than that, if $p$ is a prime number ($2, 3, 5, 7, \dots$) then a rational number has a count of how many factors of $p$ it has. Factors appearing in the denominator are counted negatively. This count is denoted $v_p(x)$.

For instance, $24/5 = 2^3 \cdot 3^1 \cdot 5^{-1}$. So we say that $v_2(24/5) = 3, v_3(24/5) = 1$ and $v_5(24/5) = -1$. We can also extend this to algebraic numbers like square roots or cube roots. For instance, $v_5(5^{1/3}) = \frac13$.

An arithmetic like the rational numbers, together with a prime-factor-counting operator like $v_5$, is called a valued field and the operator, $v_5$, is called a valuation.

Take a curve or surface defined by an algebraic equation, such as the circle $x^2 + y^2 = 1$. Look at not just the rational solutions but also the algebraic solutions. For instance, $(x, y) = (\frac15, \frac{\sqrt{24}}5)$. Consider $(v_5(x), v_5(y))$ for each algebraic solution. Here $v_5(\frac15) = -1$ and $v_5(\frac{\sqrt{24}}5) = -1$.

It turns out that the valuations for this circle are one of three kinds:

  1. $(a, a)$ for $a \le 0$
  2. $(a, 0)$ for $a \ge 0$
  3. $(0, a)$ for $a \ge 0$
graph of the tropical curve defined by the previous sets of points
Tropical Curve

Algebraic geometry is the study of these curves and surfaces defined by algebraic equations. Tropical geometry is the study of these tropical curves which are a collection of lines, rays, line segments (and the equivalent for surfaces) that arise from this process of looking at valuations.

Resources

Tropical Geometry

For a quick explanation of tropical geometry, Madeline Brandt has an 8 minute video explanation on YouTube. For a longer, but still quite approachable, introduction to tropical geometry, I recommend this survey article by Ralph Morrison. For higher level general treatments of the subject, here are some books:

Hyperfields

For learning about hyperfields and multiplicities for polynomials over hyperfields, the original article Descartes' rule of signs, Newton polygons, and polynomials over hyperfields by my PhD advisor, Matt Baker, and Oliver Lorscheid should be accessible to anyone with some familiarity with abstract algebra and mathematical maturity.

Ordered Blueprints

Oliver Lorscheid naturally has several papers describing ordered blueprints. For a starting point, I recommend looking at his lecture notes. I strongly suggest having good familiarity with commutative algebra and some category theory for these.