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Rationality of motivic Chow series modulo A^1-homotopy

Consider the formal power series $\sum [C_{p, α}(X)]t^α$ (called Motivic Chow Series), where $C_p(X)=\disjoint C_{p, α}(X)$ is the Chow variety of $X$ parametrizing the $p$-dimensional effective cycles on $X$ with $C_{p, α}(X)$ its connected components, and $[C_{p, α}(X)]$ its class in $K(ChM)_{A^1}$, the $K$-ring of Chow motives modulo $A^1$ homotopy. Using Picard product formula and Torus action, we will show that the Motivic Chow Series is rational in many cases. We have added the computation of the motivic zeta series in some of our examples so the reader can compare both series in each case.