## Segre's theorem and the MDS conjecture

#### Simeon BallPolytechnical University of Cataluña (Barcelona)Matematica Aplicada IV

Let F be a finite field. In 1955 Beniamino Segre proved that if the characteristic of F is odd then a set of |F|+1 points in the plane over F in general position (i.e. any three span the plane) is a conic. In this talk I will aim to explain how to prove the following generalization. If the characteristic of F is greater than the dimension k then a set of |F|+1 points in k-dimensional space over F in general position (i.e. any k+1 span the whole space) is a normal rational curve. In the case that |F| is prime this proves the MDS conjecture, which states that if the dimension is less than |F| then |F|+2 points cannot be in general position unless the dimension is 2 or |F|-2 and |F| is even. The MDS conjecture remains open for non-prime finite fields.

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