  Rational coefficients of special functions in scattering amplitudes are known
to simplify on singular surfaces, often diverging less strongly than the
na\"ive expectation. To systematically study these surfaces and rational
functions on them, we employ tools from algebraic geometry. We show how the
divergences of a rational function constrain its numerator to belong to
symbolic powers of ideals associated to the singular surfaces. To study the
divergences of the coefficients, we make use of $p$-adic numbers, closely
related to finite fields. These allow us to perform numerical evaluations close
to the singular surfaces in a stable manner and thereby characterize the
divergences of the coefficients. We then use this information to construct
low-dimensional Ans\"atze for the rational coefficients. As a proof-of-concept
application of our algorithm, we reconstruct the two-loop $0 \rightarrow q\bar
q\gamma\gamma\gamma$ pentagon-function coefficients with fewer than 1000
numerical evaluations.
