### Abstract

We consider the linear vector space formed by the elements of the finite field F_{q} with q = p^{r} over F_{p}. Then the elements x of F_{q} have a unique representation in the form x = Σ^{r}_{j=1} c_{j}a_{j} with c_{j} ∈ F_{p}; the coefficients c_{j} will be called digits. Let D be a subset of F_{p} with 2 ≤ |D| <p. We consider elements x of F_{q} such that for their every digit c_{j} we have c_{j} ∈ D; then we say that the elements of F_{p} \ D are "missing digits". We will show that if D is a large enough subset of F_{p}, then there are squares with missing digits in F_{q}; if the degree of the polynomial f(x) ∈ F_{q}[X] is at least 2 then it assumes values with missing digits; there are generators g in F_{q} such that f(g) is of missing digits.

Original language | English |
---|---|

Pages (from-to) | 65-74 |

Number of pages | 10 |

Journal | Functiones et Approximatio, Commentarii Mathematici |

Volume | 52 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jul 1 2015 |

### Fingerprint

### Keywords

- Character sums
- Digits properties
- Finite fields
- Generators
- Polynomials
- Primitive roots
- Squares

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Functiones et Approximatio, Commentarii Mathematici*,

*52*(1), 65-74. https://doi.org/10.7169/facm/2015.52.1.5

**Polynomial values and generators with missing digits in finite fields.** / Dartyge, Cécile; Mauduit, Christian; Sárközy, A.

Research output: Contribution to journal › Article

*Functiones et Approximatio, Commentarii Mathematici*, vol. 52, no. 1, pp. 65-74. https://doi.org/10.7169/facm/2015.52.1.5

}

TY - JOUR

T1 - Polynomial values and generators with missing digits in finite fields

AU - Dartyge, Cécile

AU - Mauduit, Christian

AU - Sárközy, A.

PY - 2015/7/1

Y1 - 2015/7/1

N2 - We consider the linear vector space formed by the elements of the finite field Fq with q = pr over Fp. Then the elements x of Fq have a unique representation in the form x = Σrj=1 cjaj with cj ∈ Fp; the coefficients cj will be called digits. Let D be a subset of Fp with 2 ≤ |D| q such that for their every digit cj we have cj ∈ D; then we say that the elements of Fp \ D are "missing digits". We will show that if D is a large enough subset of Fp, then there are squares with missing digits in Fq; if the degree of the polynomial f(x) ∈ Fq[X] is at least 2 then it assumes values with missing digits; there are generators g in Fq such that f(g) is of missing digits.

AB - We consider the linear vector space formed by the elements of the finite field Fq with q = pr over Fp. Then the elements x of Fq have a unique representation in the form x = Σrj=1 cjaj with cj ∈ Fp; the coefficients cj will be called digits. Let D be a subset of Fp with 2 ≤ |D| q such that for their every digit cj we have cj ∈ D; then we say that the elements of Fp \ D are "missing digits". We will show that if D is a large enough subset of Fp, then there are squares with missing digits in Fq; if the degree of the polynomial f(x) ∈ Fq[X] is at least 2 then it assumes values with missing digits; there are generators g in Fq such that f(g) is of missing digits.

KW - Character sums

KW - Digits properties

KW - Finite fields

KW - Generators

KW - Polynomials

KW - Primitive roots

KW - Squares

UR - http://www.scopus.com/inward/record.url?scp=84934344095&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84934344095&partnerID=8YFLogxK

U2 - 10.7169/facm/2015.52.1.5

DO - 10.7169/facm/2015.52.1.5

M3 - Article

VL - 52

SP - 65

EP - 74

JO - Functiones et Approximatio, Commentarii Mathematici

JF - Functiones et Approximatio, Commentarii Mathematici

SN - 0208-6573

IS - 1

ER -