Geodesic
Note on the Number of Solutions of f(x1) = f(x2) = ... =f(xr) over a Finite Field
Canadian mathematical bulletin, Tome 14 (1971) no. 3, pp. 429-432

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Let GF(q) denote the finite field with q = pn elements and let (1) where each ai ∊ GF(q) and 1 < d <p. For r=2, 3, ..., d we let nr denote the number of solutions (x1, ..., xr) over GF(q) of (2) for which x1, ..., xr are all different. Birch and Swinnerton-Dyer [1] have shown that (3) where each vr is a nonnegative integer depending on f and q and the constant implied by the O-symbol depends here, and throughout the paper, only on d. 
Williams, Kenneth S. Note on the Number of Solutions of f(x1) = f(x2) = ... =f(xr) over a Finite Field. Canadian mathematical bulletin, Tome 14 (1971) no. 3, pp. 429-432. doi: 10.4153/CMB-1971-074-0
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     title = {Note on the {Number} of {Solutions} of f(x1) = f(x2) = ... =f(xr) over a {Finite} {Field}},
     journal = {Canadian mathematical bulletin},
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     year = {1971},
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     doi = {10.4153/CMB-1971-074-0},
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