We devise a new criterion for linear independence over function fields. Using this tool in the setting of dual t-motives, we find that all algebraic relations among special values of the geometric Γ-function over F q [T ] are explained by the standard functional equations.

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Let k be a field of zero characteristic, and let F be a function field over k of genus g. We normalize each valuation v on F so that its order group consists of all rational integers, and for elements u1, …, un of F, not all zero, we define the (projective) height asThe sum formula on F shows that this is really a height on the projective space .

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Abstract. We investigate the arithmetic nature of special values of Thakur's function field Gamma function at rational points. Our main result is that all linear dependence relations over the field of algebraic functions are consequences of the Anderson-Deligne-Thakur bracket relations.