The strength of a homogeneous polynomial $f$ is the least integer $r$ so that $f$ can be written as a sum of $r$ reducible polynomials, $\displaystyle f = \sum_{i=1}^r g_i h_i$. This invariant has appeared several times in different contexts in math: first in analytic number theory, introduced by Schmidt in the 80s under the name ‘‘h-rank’’, then in 2009 in work by Green and Tao on higher-order Fourier analysis over finite fields, and more recently in 2018 in Ananyan and Hochster’s proof of Stillman’s conjecture in commutative algebra. Strength is also related to tensor decompositions, since it is a symmetrized version of what tensor researchers call ‘‘partition rank.’’

Most research activity involving strength has taken place when the base field is algebraically closed. My research examines what behaviors can arise over non-algebraically closed fields of interest to number theorists, such as $\mathbb{Q}, \mathbb{R}$, and $\mathbb{Q}_p$. We can view a polynomial having low strength as a generalization of having a rational point. If $f$ is a polynomial in $n+1$ variables over a field $k$, and it has a $k$-rational point, then its strength is at most $n$ (while in general it could be as high as $n+1$). Thus, saying that $f$ has less than maximal strength is a generalization of saying that $f$ has a rational point. There’s more to this connection: in analytic number theory, bounds on the strength of a polynomial can be used as input to the circle method, which allow one to obtain good estimates for the number of rational points.

There are many questions in the world of rational points that make sense to ask about strength. Can we count how many polynomials of bounded naive height over $\mathbb{Z}$ have prescribed strength? Can we count how many polynomials have prescribed strength over a finite field? Can we determine $p$-adic densities for polynomials of prescribed strength over $\mathbb{Q}_p$? Is there a local-global principle, that is, to what extent is the strength of $f$ over a global field determined by its strength over completions of that global field?

Work in progress

Arithmetic strength of pencils of conics. For two quadratics $f_1,f_2 \in k[x,y,z]$, the collective strength $\operatorname{str}(f_1,f_2) \in {1,2}$. For $k = \mathbb{R}, \mathbb{Q}, \mathbb{F}_q, \mathbb{Q}_p$, and finite extensions of $\mathbb{Q}_p$, I investigate the question ‘‘how often is $\str_k(f_1,f_2)$ less than maximal?’’ For $\mathbb{R}$, the answer is ‘‘always’’, for $\mathbb{Q}$ the answer is ‘‘almost never’’, but for $\mathbb{F}_q$ and extensions of $\mathbb{Q}_p$ it is a proportion between 0 and 1, tending to $\frac{5}{12}$ as the residue characteristic tends to infinity.

Undergraduate publications

Gonality sequences of graphs with Franny Dean, Ralph Morrison, Teresa Yu, and Julie Yuan, SIAM Journal on Discrete Mathematics 35 (2021), no. 2, 814–839. (arxiv)
On the gonality of Cartesian product graphs with Ralph Morrison, Electronic Journal of Combinatorics 27 (2020), no. 4, Paper No. 4.52, 35 pp. (arxiv)
Graphs of gonality 3 with Franny Dean, Ralph Morrison, Teresa Yu, and Julie Yuan, Algebraic Combinatorics, Volume 2 (2019) no. 6 p. 1197-1217. (arxiv)
Treewidth and gonality of glued-grid graphs with Franny Dean, Ralph Morrison, Teresa Yu, and Julie Yuan, Discrete Applied Mathematics 279 (2020), 1-11. (arxiv)