  Intersection numbers are rational scalar products among functions that admit
suitable integral representations, such as Feynman integrals. Using these
scalar products, the decomposition of Feynman integrals into a basis of
linearly independent master integrals is reduced to a projection. We present a
new method for computing intersection numbers that only uses rational
operations and does not require any integral transformation or change of basis.
We achieve this by systematically employing the polynomial series expansion,
namely the expansion of functions in powers of a polynomial. We also introduce
a new prescription for choosing dual integrals, de facto removing the explicit
dependence on additional analytic regulators in the computation of intersection
numbers. We describe a proof-of-concept implementation of the algorithm over
finite fields and its application to the decomposition of Feynman integrals at
one and two loops.
