Scattering amplitudes in perturbative quantum field theory exhibit a rich structure of zeros, poles and branch cuts which are best understood in complexified momentum space. It has been recently shown that by leveraging this information one can significantly simplify both analytical reconstruction and final expressions for the rational coefficients of transcendental functions appearing in phenomenologically-relevant scattering amplitudes. Inspired by these observations, we present a new algorithmic approach to the reconstruction problem based on p-adic numbers and computational algebraic geometry. For the first time, we systematically identify and classify the relevant irreducible surfaces in spinor space with five-point kinematics, and thanks to p-adic numbers – analogous to finite fields, but with a richer structure to their absolute value – we stably perform numerical evaluations close to these singular surfaces, thus completely avoiding the use of floating-point numbers. Then, we use the data thus acquired to build ansätze which respect the vanishing behavior of the numerator polynomials on the irreducible surfaces. These ansätze have fewer free parameters, and therefore reduced numerical sampling requirements. We envisage future applications to novel two-loop amplitudes.