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The kinetics of dendritic solidification in a ternary Ni₉₈Zr₁Al₁ alloy is investigated experimentally in a range of melt undercoolings 40 K ≤ ΔT≤ 320 K. The growth velocity is measured for samples processed by the electromagnetic levitation technique using a high-speed video camera. With ΔT ≤ 220 K the measured growth rates are the same as those of a binary Ni₉₉Zr₁ alloy. In the regime of rapid solidification, especially within the regime of thermal dendritic growth at ΔT ≥ 220 K, growth rates are decreased. Sharp-interface modeling predicts growth rates over the whole range of undercooling. Phase-field simulations give quantitative predictions for the dendritic growth velocity in the solute-controlled growth regime. Results show that the composition and temperature dependency of the thermodynamic data, e.g. liquidus slope and solute partition coefficient, are important for describing the alloys. Our findings give improved sharp-interface model predictions compared to calculations based on an approximation of the thermodynamic data derived from binary phase diagrams-