| Full text | |
| Author(s): |
Reis, Lucas
Total Authors: 1
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| Document type: | Journal article |
| Source: | FINITE FIELDS AND THEIR APPLICATIONS; v. 68, p. 9-pg., 2020-12-01. |
| Abstract | |
Let q be a prime power, let F-q be the finite field with q elements and let d(1), ...,d(k) be positive integers. In this note we explore the number of solutions (z(1), ...,z(k)) is an element of (F) over bar (k)(q) of the equation L-1(x(1)) + ... + L-k(x(k)) = b, with the restrictions z(i) is an element of F-qdi, where each L-i (x) is a non zero polynomial of the form Sigma(mi)(j=0) a(ij)x(qj) is an element of F-q [x] and b is an element of (F) over bar (q). We characterize the elements b for which the equation above has a solution and, in affirmative case, we determine the exact number of solutions. As an application of our main result, we obtain the cardinality of the sumset Sigma(k)(i=1) F-qdi := {alpha(1) + ... + alpha(k) vertical bar alpha(i) is an element of F-qdi}. Our approach also allows us to solve another interesting problem, regarding the existence and number of elements in F-qn with prescribed traces over intermediate F-q-extensions of F-qn. (C) 2020 Elsevier Inc. All rights reserved. (AU) | |
| FAPESP's process: | 18/03038-2 - Polynomial maps in finite fields and their applications |
| Grantee: | Lucas da Silva Reis |
| Support Opportunities: | Scholarships in Brazil - Post-Doctoral |