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Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales

versión impresa ISSN 0370-3908

Rev. acad. colomb. cienc. exact. fis. nat. vol.36 no.139 Bogotá abr./jun. 2012

 

MATEMÁTICAS

ORTHOGONAL SYSTEMS AND PERMUTATION POLYNOMIAL VECTORS OVER MODULAR ALGEGRAS

Pablo A. Acosta-Solarte*, Víctor S. Albis**

* Universidad Distrital Francisco José de Caldas, Bogotá, Colombia. Email: paacostas@udistrital.edu.co
**Universidad Nacional de Colombia, Bogotá, Colombia. Email: valbis@accefyn.org.co
AMS Classification 2010: 13B25, 13F25, 11T55.


Abstract

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


Resumen

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.


1. Introduction

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 ([1] and [2]) 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, [5],[7],[8],[10],[13] and [14]). Permutation polynomial, find applications currently in cryptography and coding theory (see [4] 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 [6] when the coefficients of the polynomials are in finite fields. Moreover, WEI & ZHANG in [12] and SHIUE, SUN & ZHANG in [8] extended some of these results to certain finite rings.

2. Preliminaries.

In this section we recall some properties of and needed for the best undertading of what follows (see [3],[9]. 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

and
is the canonical projection.

If , its reduction modulo is the polynomial in whose coefficients are the classes modulo of the coefficients of . Clearly, if ≤ µ,

If

is a zero of and ≤ µ, we say that is a descendant of if ; obviously, if that is the case, = 0, and we also say that is an ascendantof . Moreover, if is a zero of , then in , ≤ µ, has ot most descendants, if any.

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 [7] introduces the notion of permutation polynomial modulo α and also the notion of regular polynomial if R/α is a finite set. In [1] 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 ([11],[12]) 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 in

On 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 [6].

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 a unique solution. Therefore the equation has at least one solution in . Let . 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

where

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 solutions, thus
is equivalent to (8). Since for some is a unit, then (9) has solutions, i.e.,

has at least solutions. Now, the polynomial is a permutation polynomial, and for in , the equation = has qv(k-1) solutions.
So, the equation


has solutions.

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 .

Proof. If


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 [6], 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. But is an orthogonal system over and, therefore,


has solutions, i.e,

is a permutation polynomial over .
Proposition 3.7. Let .
If
is a permutation polynomial over for all permutation polynomial over , , and the zeroes of are non singular for all and then is an orthogonal system over .

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,…,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.

Acknowledgements

We wish to express our thanks to Yuguang Lu for his help in the reading and understanding of [8], [10] and [14].

References

[1] Acosta, S. P. A. & Albis, V.S. Characterization of multivariate permutation polynomials in positive characteristic, Sao Paulo J. Math. Sci. 3 N. 1 (2009), 1-12.         [ Links ]

[2] Acosta, S. P. A. & Albis, V.S. Permutation polynomials in one indeterminate over modular algebras, Rev. Acad. Colomb. Cienc. 30 Nº. 117 (2006), 541-548. [MR:2334082]         [ Links ]

[3] Albis, V. S. & Chaparro, R. On a conjecture of Borevich and Shafarevich ,Rev. Acad. Colomb. Cienc. 21 (1997), 313-319. [MR:98g:11130]         [ Links ].

[4] Laigle-Chapuy, Y. Permutations polynomials and applications to coding theory., Finite Fields Appl. 13 (2007), 58-70.         [ Links ]

[5] Niederreiter, H. Permutation polynomials in finite fields, Proc. Japan Acad. 46 Nº. 9 (1970), 1001-1005. [MR: 44#5298]         [ Links ].

[6] Niederreiter, H. Orthogonal system of polynomials in finite fields, Proc. Amer. Math. Soc. 28 (1971), 415-422. [MR: 45#230]         [ Links ].

[7] Nöbauer, Wilfriend. Zur Theorie der polynomialsformationen und permutations polynome, Math. Annalen 157 (1964), 332-342.         [ Links ]

[8] Shiue, P. J. S; Sun, Q. & Zhang, Q. Multivariate permutation polynomials and orthogonal systems over residue class rings, Chinese. Ann. Math. Ser. A. 17 Nº. 1 (1996), 43-46. [in Chinese] [MR: 97e:11152]         [ Links ].

[9] Smits, T. H. . On the group of units of . Indag. Math. 44 (1982), 355-358.         [ Links ]

[10] Sun, Q. A note on permutation polynomials vectors over Adv. Math. (china) 25 Nº. 1 (1996), 311-314. . [in Chinese] [MR: 98h:11157]         [ Links ].

[11] Zhang, Q Polynomials functions and permutation polynomials over some finite commutative rings. J. Number Theory 105 (2004), 192- 202.         [ Links ]

[12] Wei, Q. & Zhang, Q. On strong orthogonal systems and weak permutation polynomials over finite commutative rings, Finite Fields Appl. 13 (2007), 113-120.         [ Links ]

[13] Zhang, Q. On the polynomials in several indeterminates which can be extended to permutation polynomial vector over , Adv, Math.22 Nº.5 (1993), 456- 457.         [ Links ]

[14] Zhang, Q. Permutation polynomial in several indeterminates over , Chinese An. Math. Ser. A. 16 Nº.2 (1995), 168-172. . [in Chinese] [MR: 96g:11143]         [ Links ].

Recibido: 5 de marzo de 2012

Aceptado para publicación: 19 de abril de 2012