相关论文: The Slope Polynomial and Collinear Points in Permu…

This paper has been withdrawn by the author due to a serious mistake on Lemma 2.4.

This paper has been withdrawn due to crucial erros in equations (24)--(31).

Let $\alpha:\mathbb{F}_q\to\mathbb{F}_q$ be a permutation and $\Psi(\alpha)$ be the number of collinear triples in the graph of $\alpha$, where $\mathbb{F}_q$ denotes a finite field of $q$ elements. When $q$ is odd Cooper and Solymosi once…

This paper has been withdrawn by the author due to a crucial errors.

This paper has been withdrawn by the author due to serious flaws in certain proofs. For instance, the method used to construct certain automorphic representations is flawed.

This paper has been withdrawn by the authors due to its publication

This paper has been withdrawn because the result turns out to be trivial.

The paper has been withdrawn due to a crucial error in section 3.

%auto-ignore This paper has been withdrawn by the author, due to a crucial error.

This paper has been withdrawn by the authors due to some fatal errors in the analysis.

The paper has been withdrawn

This paper has been withdrawn due to a crucial theoretical and experimental error.

This paper has been withdrawn due to a crucial theoretical error.

The paper is withdrawn due to mistakes in the proofs for Proposition 1.2 and Theorem 2.2.

This paper has been withdrawn by the author due to a crucial sign error in equation 1.

This paper has been withdrawn due to a crucial error in the proof of the main theorem

This paper has been withdrawn due a crucial mistake due to a crucial mistake in the proof of Lemma 2.3.

This paper has been withdrawn since it contains some discrepancy with othe authers's recent result. We will not post this until this discrepancy is resolved.

This paper has been withdrawn by the authors due to its publication

This paper has been withdrawn by the author due to a crucial error in the last lines in the proof of Lemma 3.3.