Constructing Compact Ans\"atze for Scattering Amplitudes

In these proceedings, we discuss the recent approach of Ref. [1] for the construction of compact Ans\"atze for scattering amplitudes. The method builds powerful constraints on the analytic structure of the rational functions in amplitudes from numerical tests of their behavior close to singularity surfaces. We discuss how we systematically understand these surfaces and how the singular behavior of the rational function can be incorporated into an Ansatz using techniques from algebraic geometry. To perform the numerical sampling, we make use of $p$-adic numbers, a number-theoretical field that can be considered a cousin of finite fields. The $p$-adic numbers admit a non-trivial absolute value, as well as analytic functions such as the $p$-adic logarithm. We provide a detailed example of the approach applied to an NMHV tree amplitude and discuss the efficacy when applied to the two-loop leading-color amplitude for three-photon production at hadron colliders.