Trace of products in finite fields
Copyright policy )Trace of products in finite fields
Let p be a prime number and let q = p(r). If C and D are large subsets of F-q(*) we study the trace of products cd with c is an element of C and d is an element of D and show that it is well distributed in F-p. We give an optimal condition (up to an absolute constant factor) on the size of the subsets C and D to ensure that the trace of products cd takes any given value in Fp. We also give a condition (optimal up to an absolute constant factor in most cases) on the size of the subsets C and D to ensure that the trace of cd meets the set of k-th powers for k >= 1, respectively the set of generators. Our method will enable us to take sets C and D whose size is substantially below root q. Character sums and Gaussian sums over F-p and F-q will play an important role in the proofs. Some estimates lead to interesting combinatorial questions in finite fields.
Character sums, Squares, character sums, Structure theory for finite fields and commutative rings (number-theoretic aspects), Generators, Direct product of subsets, Trace function, direct product of subsets, Finite fields, Exponential sums, finite fields, Radix representation; digital problems, squares, generators, trace function, [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]
Character sums, Squares, character sums, Structure theory for finite fields and commutative rings (number-theoretic aspects), Generators, Direct product of subsets, Trace function, direct product of subsets, Finite fields, Exponential sums, finite fields, Radix representation; digital problems, squares, generators, trace function, [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]
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