versión impresa ISSN 0370-3908
Rev. acad. colomb. cienc. exact. fis. nat. vol.36 no.139 Bogotá abr./jun. 2012
* Universidad Distrital Francisco José de Caldas, Bogotá, Colombia. Email: email@example.com
**Universidad Nacional de Colombia, Bogotá, Colombia. Email: firstname.lastname@example.org
AMS Classification 2010: 13B25, 13F25, 11T55.
Known results on orthogonal systems and permutation polynomials vectors over finite fields are extended to modular algebras of the form , where K is a finite field, is an irreducible polynomial, = 1, 2, . . ., and to the algebra of formal power series , where L1 = K[X]/(p(X)) = L.
Key words: Permutation polynomial, orthogonal systems, permutation polynomial vectors
Resultados sobre sistemas ortogonales y vectores de polinomios de permutación se extienden a las álgebras modulares de la forma , donde K es un cuerpo finito, un polinomio irreducible, = 1, 2, . . . y al álgebra de las series potenciales formales , donde L1 = K[X]/(p(X)) = L.
Palabras clave: Polinomio de permutación, sistemas ortogonales, vectores de polinomios de permutación.
Let be a finite field, its ring of polynomials and an irreducible monic polynomial. It is known that is a finite field an that , = 1,2,..., are L-algebras (see infra for details). In previous papers ( and ) the authors obtained results about permutation polynomial over the L-algebras (formal series over L) and , analogos to some known results over finite fields, Galois rings , and the rings (see, for example, ,,,, and ). Permutation polynomial, find applications currently in cryptography and coding theory (see  for more references).
In this paper we deal with systems of polynomials in and , obtaining results than in some cases lead to construct new permutation polynomials. The systems we are interested in are know as orthogonal systems has being studied by NIEDERREITER in  when the coefficients of the polynomials are in finite fields. Moreover, WEI & ZHANG in  and SHIUE, SUN & ZHANG in  extended some of these results to certain finite rings.
In this section we recall some properties of and needed for the best undertading of what follows (see ,. Here the elements of will be denoted by
wher is the class of equivalence modulo The elements of L will simply be denote by α It is know that
is a local ring with maximal ideal (Z), and
are the only ideals of . Also, , and is a finite ring with elements, (=1,2,...,) when L has q elements. Thus is the projective limit of the projective system of L-algebras where
If , its reduction modulo is the polynomial in whose coefficients are the classes modulo of the coefficients of . Clearly, if ≤ µ,
A zero ∈ of is said to be non-singular if
for some j= 1,...,n. Otherwise is called a singular zero. It is clear that any descendant (resp. ascendant) of a non-singular zero is a non-singular zero.
3. Orthogonal systems and permutation polynomial vectors.In this section we introduce definitions and some results on the systems we are interested in. For a given commutative ring , and , and α an ideal of R, W. NÖBAUER  introduces the notion of permutation polynomial modulo α and also the notion of regular polynomial if R/α is a finite set. In  we proved that a permutation polynomial modulo is also a regular polynomial in . More precisely, we prove that induces a permutation polynomial over if, and only if, the equation has exactly solutions for each
Accordingly to ZHANG (,) this means that the polynomial induces a permutation polynomial over if, and only if, is a uniform map.
The system of polynomials in is said to be an orthogonal system over if the map given by for is a uniform map over , i.e., if the system of equations
has solutions in where is the reduction of in . If n = k the system is called permutation polynomial vector (PPV) over
Is clear that when k = 1, an orthogonal system is a permutation polynomial.
Proposition 3.1. Let . If is an orthogonal system over the it is and orthogonal system over In particular, it is an orthogonal system over L.
Proof. Let be an orthogonal system over . Then the system
has solution in . Let N be the number of these solutions. Each of them has descendants. On the other hand, from (2) we see that there are systems of the form
each of which has, by hypothesis, different solutions, i.e., taken altogether all the above systems will have different solutions. Since each solution descends from then
therefore, . So, is an orthogonal system over .
Corollary 3.1. Let be a permutation polynomial vector over , then is a permutation polynomial vector over . In particular is a permutation polynomial vector over L.
Proposition 3.2. let be an orthogonal system over L1 = L, and such that the zeroes of are nonsingular for all . Then is an orthogonal system over (= 1,2,...).Proof. Let N be the number of solutions of (1), where . We obtain the system
which has solutions. Since the zeroes of are non singular, each of the polynomial in (3) has exactly descendants in ([1, lem. 2.2 ]). All of them are not different, since otherwise each zero of (4) would have descendants and since each element of L can be viewed in ways in , then for k > 1, a zero of (4) would have more than ways to be viewed inOn the other hand, if k = 1 then the proposition is true by proposition 3.1 in ((1)). Now, if these descendants were the same for each polynomial in (4), then (3) would have different solutions. Then the system would have
different solutions, thus for > 1, k >1. But this contradicts, the cardinality of .
Therefore, the number of descendants, let us say D, contributed, by each polynomial in the system (4) to the solutions of system (3) is such that
Thus, and the total number of solutions of (3) is is an orthogonal system over
Corollary 3.2. Let be a PPV over L such that the zeroes of are nonsingular for all . Then is a PPV over .The following propositions are extensions to L[[Z]] of results known in finite fields, see .
Proposition 3.3. Let > 1 be, and be polynomials such that is a PPV over then, the projection of in is a permutation polynomial.
Proof. If is a PPV over then, for all the system
has again a unique solution, different to the solution of (5), because, otherwise,. But this can be done in ways, i.e., for has at least solutions. If there is one more solution, say we can construct the system
system that necessarily is one of previous system (6). Therefore has exactly solutions, thus is a permutation polynomial.
Corollary 3.3. Every polynomial in a PPV is a permutation polynomial.
Proposition 3.4. Let be an orthogonal system over and and at least one them a unit.
Then the polynomial
is a permutation polynomial over
Proof. Let be . We see that the number of solutions of
is . By hypothesis, the system,
has at least solutions. Now, the polynomial is a permutation polynomial, and for in , the equation = has qv(k-1) solutions.
So, the equation
Corollary 3.4. Every polynomial an orthogonal system is a permutation polynomial.
Proposition 3.5. If
is a permutation polynomial over and the zeroes of are non singular for all and where at least one them is a unit, then is an orthogonal system over .
is a permutation polynomial then is also a permutation polynomial ([1, lem. 3.3]). By hypothesis, the zeroes of are non singular; then by the corollary to theorem 2 in , the system is an orthogonal system over L and by proposition 3.2 is an orthogonal system over .
Proposition 3.6. Let . If is an orthogonal system over then for all permutation polynomial over , the polynomial
is a permutation polynomial.
Proof. Let be . Since is a permutation polynomial, has solutions in and the system
has solutions, i.e,
is a permutation polynomial over .
Proposition 3.7. Let .
Proof. Since is a permutation polynomial, for all permutation polynomial, for all permutation polynomial , in particular it is a permutation polynomial for , where at least one is a unit. Then by proposition 3.5, the system is an orthogonal system over .
Proposition 3.8. Let be a polynomials system. Then there exist coefficients , where at least one of them is a unit, such that
is not a permutation polynomial.
Proof. Let where at least one of them is a unit. If the polynomial
were a permutation polynomial, then the polynomial
is also a permutation polynomial with different from (0,â¦,0). This contradicts [6, theor. 4].
Proposition 3.9. If is an orthogonal system, then any of its nonempty subsystem is again an orthogonal system.
Proof. If an orthogonal system then
has solutions. The lemma is proved, without lost of generality, if the system
has solutions. Then, for all the equation (11) has at least solutions, the same as (10). If we take , then again for this (10) has solutions, which are different to the initial ones; therefore for each , the equation (11) has solutions more. In total (11) has solutions.
Corollary 3.5. If is a PPV, then any of its nonempty subsystems is an orthogonal system.
Proof. It is clear from proposition 3.9 and the definition of PPV.
We wish to express our thanks to Yuguang Lu for his help in the reading and understanding of ,  and .
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Recibido: 5 de marzo de 2012
Aceptado para publicación: 19 de abril de 2012