Abstract: In this paper, we consider some polynomial (α,β)-metrics, and discuss the sufficient and necessary conditions for a Finsler metric in the form F=α+a₁β+a₂β²/α+a₄β⁴/α³ to be projectively flat, where a_i (i=1,2,4) are constants with a₁≠0, α is a Riemannian metric and β is a 1-form. By analyzing the geodesic coefficients and the divisibility of certain polynomials, we obtain that there are only five projectively flat cases for metrics of this type. This gives a classification for such kind of Finsler metrics.

Key words: Finsler metric, Polynomial, Projectively flat

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