Given any graded polynomial ring over an algebraically closed field, we define the projective toric variety cut out by relations between monomials of a fixed degree. We specifically study the case of the free commutative algebra generated by Lyndon words in letters 2 and 3, where a Lyndon word is assigned the weight (degree) given by the sum of its digits, e.g. wt(23233)=13. Due to a result by Francis Brown, it is known that the algebra of multiple Zeta values, a generalisation of positive integer Riemann Zeta values, is generated by the values for precisely the Lyndon words in 2 and 3. Conjecturally, it is isomorphic to the polynomial ring in these Lyndon words. Thus, our toric ideals describe all the _expected_ _nonlinear_ relations between multiple zeta values of a _fixed_ weight.

Joint work with Annika Burmester, Steven Charlton, Abhiram Kidambi and Felix Lotter