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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mo stretchy="false">(</a:mo> <a:mi>n</a:mi> <a:mo>−</a:mo> <a:mn>1</a:mn> <a:mo stretchy="false">)</a:mo> <a:mo>!</a:mo> <a:mo>/</a:mo> <a:mn>2</a:mn> <a:mo>−</a:mo> <a:mn>2</a:mn> </a:math> relations for <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>n</e:mi> <e:mo>≥</e:mo> <e:mn>5</e:mn> </e:math> , 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 <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>n</g:mi> <g:mo>=</g:mo> <g:mn>7</g:mn> </g:math> . We also study relations for the tree-level maximally helicity violating amplitudes through <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>n</i:mi> <i:mo>=</i:mo> <i:mn>8</i:mn> </i:math> . The latter relations include the well-known color and Bern-Carrasco-Johansson identities.