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The leptonic decay-constant ratio $f_K/f_\pi$ is calculated from lattice-QCDsimulations using $N_f=2+1$ dynamical fermion flavors in the clover-improvedformulation and 2-HEX smearing. The simulations were performed for a range ofmass-degenerate light quarks including the physical point and at variouslattice couplings and volumes, allowing to quantify all relevant sources ofsystematic uncertainties for our final number of the decay-constant ratio.Utilizing input from chiral perturbation theory, we also quote the chargeddecay-constant ratio $f_{K^\pm}/f_{\pi^\pm}$. With further input fromsuper-allowed nuclear $\beta$-decays, eventually we obtain an estimate for theCKM-matrix element $V_{\rm us}$.

Leptonic decay-constant ratio $f_K/f_\pi$ from clover-improved $N_f=2+1$ QCD