Sharp symmetry bounds for real plane algebraic curves (Theorem 1)
Let C be an infinite real plane algebraic curve defined by a polynomial of degree d >= 2 that is irreducible over the complex numbers, and suppose C is not a circle. Its group of Euclidean symmetries is finite, the orientation-preserving part is cyclic of order at most max(d, 2d - 4), and the full group has order at most 2d. Both bounds are attained in every degree d >= 2. For d >= 5, curves attaining the rotation bound are exactly, up to orientation-preserving similarity, the curves Re(z^(d-2)(|z|^2 + a)) = 0 with |a| = 1 and a not real; conversely each such curve with d - 2 >= 3 attains…
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