Serendipitous syzygies of scattering amplitudes

We study linear relations between color-ordered all-plus amplitudes at one loop in Yang-Mills theory. We show that on general grounds, there are ( n − 1 ) ! / 2 − 2 relations for n ≥ 5 , leaving only two independent color-ordered amplitudes. We present two complementary approaches to finding such relations; one using numerical linear algebra and the other using syzygies in computational algebraic geometry. We obtain explicit forms for all relations through n = 7 . We also study relations for the tree-level maximally helicity violating amplitudes through n = 8 . The latter relations include the well-known color and Bern-Carrasco-Johansson identities.