Algebraic geometry and p-adic numbers for scattering amplitude ansätze

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