# On the cylinder conjecture

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## Abstract

In this paper, we associate a weight function with a set of points satisfying the conditions of the cylinder conjecture. Then we derive properties of this weight function using the Rédei polynomial of the point set. Using additional assumptions on the number of non-determined directions, together with an exhaustive computer search for weight functions satisfying particular properties, we prove a relaxed version of the cylinder conjecture for \(p \le 13\). This result also slightly refines a result of Sziklai on point sets in \(\mathrm {AG}(3,p)\).

## Keywords

Cylinder conjecture Polynomial method Affine space## Mathematics Subject Classification

05B25 51D20## Notes

### Acknowledgements

The authors acknowledge Aart Blokhuis and Klaus Metsch for the fruitful discussions on the combinatorial part (Sect. 2).

## References

- 1.Alon N.: Combinatorial nullstellensatz. Combin. Probab. Comput.
**8**(1–2), 7–29 (1999). Recent trends in combinatorics (Mátraháza, 1995).MathSciNetCrossRefGoogle Scholar - 2.Ball S.: On the graph of a function in many variables over a finite field. Des. Codes Cryptogr.
**47**(1–3), 159–164 (2008).MathSciNetCrossRefGoogle Scholar - 3.Ball S., Lavrauw M.: On the graph of a function in two variables over a finite field. J. Algebr. Comb.
**23**(3), 243–253 (2006).MathSciNetCrossRefGoogle Scholar - 4.Blokhuis A.: Private communication (2017).Google Scholar
- 5.Blokhuis A., Marino G., Mazzocca F.: Generalized hyperfocused arcs in \(PG(2, p)\). J. Comb. Des.
**22**(12), 506–513 (2014).MathSciNetCrossRefGoogle Scholar - 6.Sziklai P.: Directions in \({\rm AG}(3, p)\) and their applications. Note Mat.
**26**(1), 121–130 (2006).MathSciNetzbMATHGoogle Scholar

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