Generalized A=B relations

Let $A^1_{g,n}$ and $B^1_{g,n}$ be the polynomial-valued tautological classes associated with the rubber and corresponding constructions in the paper, and let $B^m_{g,n}$ and $B^0_{g,n}$ denote the analogous classes. Let

$$\pi\colon \overline{\mathcal{M}}_{g,n+1}\to\overline{\mathcal{M}}_{g,n}$$

be the morphism forgetting the last marked point. Generalized $A=B$ relations. The following statements hold: for $g\geq0$, $n\geq1$, and $m\geq2$, $\deg B^m_{g,n}\leq2g-2+m$; for $g\geq0$, $n\geq1$, and $2g-1+n>0$, $\deg(B^1_{g,n}-A^1_{g,n})\leq2g-1$; and for $g\geq0$, $n\geq1$, and $2g-2+n>0$,

$$\deg\left(B^0_{g,n}-\frac{1}{\sum_{i=1}^n a_i}\pi_*A^1_{g,n}\right)\leq2g-2.$$

These degree bounds formulate the generalized $A=B$ relations for the tautological classes; their validity is presented as conjectural in the source.