%0 Journal Article %T Polynomial values and generators with missing digits in finite fields %+ Institut Élie Cartan de Lorraine (IECL) %+ Institut de Mathématiques de Marseille (I2M) %+ Department of Algebra and Number Theory [Budapest] %A Dartyge, Cécile %A Mauduit, Christian %A Sárközy, András %< avec comité de lecture %@ 0208-6573 %J Functiones et Approximatio Commentarii Mathematici %I Poznań : Wydawnictwo Naukowe Uniwersytet im. Adama Mickiewicza %V 52 %N 1 %P 65-74 %8 2015-03 %D 2015 %R 10.7169/facm/2015.52.1.5 %K generators %K polynomials %K primitive roots %K digits properties %K finite fields %K character sums %K squares %Z Primary 11A63; Secondary 11L99 %Z Mathematics [math]/Number Theory [math.NT]Journal articles %X We consider the linear vector space formed by the elements of the finite field Fq with q =Ppr over Fp. Then the elements x of Fq have a unique representation in the form x = rj=1 cjaj with cj ∈ Fp; the coefficients cj will be called digits. Let D be a subset of Fp with 2 6 |D| < p. We consider elements x of Fq such that for their every digit cj we have cj ∈ D; then we say that the elements of Fp r 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. %G English %L hal-01272382 %U https://hal.science/hal-01272382 %~ CNRS %~ UNIV-AMU %~ IECN %~ EC-MARSEILLE %~ INSMI %~ I2M %~ I2M-2014- %~ UNIV-LORRAINE %~ IECLANALYSE %~ ANR