Gröbner Bases for Linearized Polynomials
aa r X i v : . [ c s . S C ] J un Gr¨obner Bases for Linearized Polynomials
Margreta Kuijper and Anna-Lena TrautmannDepartment of Electrical and Electronic EngineeringUniversity of Melbourne, Australia.
Abstract
In this work we develop the theory of Gr¨obner bases for modules overthe ring of univariate linearized polynomials with coefficients from a finitefield.
Gr¨obner bases [2] are a powerful conceptual and computational tool for modulesover general multivariate polynomial rings. In particular, they also prove usefulfor modules over univariate polynomials with coefficients from a finite field. Thismotivates us to develop similar tools for modules over linearized polynomials,a special family of polynomials over a finite field, in an analogous manner. Formore information on Gr¨obner bases for modules over finite field polynomial ringsthe interested reader is referred to [1].Let F q denote the finite field with q elements, where q is a prime power, andlet F q m denote the extension field of extension degree m . A linearized polynomial over F q m is of the form f ( x ) = n X i =0 a i x [ i ] , a i ∈ F q m , where [ i ] := q i . If the base field needs to be specified, these polynomials are alsocalled q -linearized. The name ”linearized” stems from the fact that linearizedpolynomials function as q -linear maps. This class of polynomials was first in-vestigated in [10] and later on by [3]. These polynomials have received a lot ofinterest in the past decades due to their application in rank-metric codes [3, 4]and related topics.The set of linearized polynomials, equipped with normal polynomial addition+ and polynomial composition ◦ , forms a non-commutative ring without zero-divisors (see e.g. [5]). We will denote this ring of q -linearized polynomials over F q m by L q ( x, q m ).Due to the difference of composition (for linearized polynomials) and multi-plication (for classical polynomials), the theory of bases in general, and Gr¨obnerbases in particular, needs to be developed from scratch for the ring L q ( x, q m ).The paper is structured as follows: In the next section we will investigatethe structure of L q ( x, q m ) ℓ as a left module. In Section 3 we will derive thetheory of Gr¨obner bases for submodules of L q ( x, q m ) ℓ . We conclude this workin Section 4. 1 The Module L q ( x, q m ) ℓ As mentioned before, L q ( x, q m ) forms a ring with addition and composition.Hence L q ( x, q m ) ℓ forms a right or left module, which are different due to thenon-commutativity of ◦ . In this work we will consider L q ( x, q m ) ℓ as a leftmodule and investigate its left submodules. The results then easily carry overto right modules.Elements of L q ( x, q m ) ℓ are of the form f := [ f ( x ) . . . f ℓ ( x )] = ℓ X i =1 f i ( x ) e i where f i ( x ) = P j f ij x [ j ] ∈ L q ( x, q m ) and e , . . . , e ℓ are the unit vectors oflength ℓ . To avoid confusion we denote polynomials by f ( x ), while vectorsof polynomials are denoted by f . If we need to index polynomials, we usethe notation f ( x ) , . . . , f s ( x ), while for vectors of polynomials we will use thenotation f (1) , . . . , f ( s ) . Analogous to polynomial multiplication on F q m [ x ] ℓ wedefine for h ( x ) ∈ L q ( x, q m ) the left operation h ( x ) ◦ f := [ h ( f ( x )) . . . h ( f ℓ ( x ))] = ℓ X i =1 h ( f i ( x )) e i . The monomials of f are of the form x [ k ] e i for all k such that f ik = 0. Definition 1.
A subset M ⊆ L q ( x, q m ) ℓ is a (left) submodule of L q ( x, q m ) ℓ ifit is closed under addition and composition with L q ( x, q m ) on the left. Definition 2.
Consider the non-zero elements f (1) , . . . , f ( s ) ∈ L q ( x, q m ) ℓ . Wesay that f (1) , . . . , f ( s ) are linearly independent if for any a ( x ) , . . . , a s ( x ) ∈L q ( x, q m ) s X i =1 a i ( x ) ◦ f ( i ) = [ 0 . . . ⇒ a ( x ) = · · · = a s ( x ) = 0 . A generating set of a submodule M ⊆ L q ( x, q m ) ℓ is called a basis of M if all itselements are linearly independent.One can easily see that B = { xe , xe . . . , xe ℓ } is a basis of L q ( x, q m ) ℓ , thus L q ( x, q m ) ℓ is a free and finitely generated module.We need the notion of monomial order for the subsequent results, which wewill define in analogy to [1, Definition 3.5.1]. Definition 3. A monomial order < on L q ( x, q m ) ℓ is a total order on L q ( x, q m ) ℓ that fulfills the following two conditions: • x [ k ] e i < x [ j ] ◦ ( x [ k ] e i ) for any monomial x [ k ] e i ∈ L q ( x, q m ) ℓ and j ∈ N > . • If x [ k ] e i < x [ k ′ ] e i ′ , then x [ j ] ◦ ( x [ k ] e i ) < x [ j ] ◦ ( x [ k ′ ] e i ′ ) for any monomials x [ k ] e i , x [ k ′ ] e i ′ ∈ L q ( x, q m ) ℓ and j ∈ N .2n example of a monomial order on L q ( x, q m ) ℓ is the weighted term-over-position monomial order in [8]. In the following we will not fix a monomialorder. The results are general and hold for any chosen monomial order. Definition 4.
We can order all monomials of an element f ∈ L q ( x, q m ) ℓ indecreasing order with respect to some monomial order. Rename them such that x [ i ] e j > x [ i ] e j > . . . . Then1. the leading monomial lm( f ) = x [ i ] e j is the greatest monomial of f .2. the leading position lpos( f ) = j is the vector coordinate of the leadingmonomial.3. the leading term lt( f ) = f j ,i x [ i ] e j is the complete term of the leadingmonomial.In order to define minimality for submodule bases we need the followingnotion of reduction, in analogy to [1, Definition 4.1.1]. Definition 5.
Let f, h ∈ L q ( x, q m ) ℓ and let F = { f (1) , . . . , f ( s ) } be a set ofnon-zero elements of L q ( x, q m ) ℓ . We say that f reduces to h modulo F in onestep if and only if h = f − (( b x [ a ] ) ◦ f (1) + · · · + ( b k x [ a k ] ) ◦ f ( k ) )for some a , . . . , a k ∈ N and b , . . . , b k ∈ F q m , wherelm( f ) = x [ a i ] ◦ lm( f ( i ) ) , i = 1 , . . . , k, andlt( f ) = ( b x [ a ] ) ◦ lt( f (1) ) + · · · + ( b k x a [ k ] ) ◦ lt( f ( k ) ) . We say that f is minimal with respect to F if it cannot be reduced modulo F . Definition 6.
A module basis B is called minimal if all its elements b areminimal with respect to B \{ b } . Proposition 7. [7] Let B be a basis of a module M ⊆ L q ( x, q m ) ℓ . Then B is a minimal basis if and only if all leading positions of the elements of B aredistinct.Proof. Let B be minimal. If two elements of B have the same leading position,the one with the greater leading monomial can be reduced modulo the otherelement, which contradicts the minimality. Hence, no two elements of a minimalbasis can have the same leading position.The other direction follows straight from the definition of reducibility andminimality of a basis, since if the leading positions of all elements are different,none of them can be reduced modulo the other elements.The property outlined in the following theorem is called the PredictableLeading Monomial (PLM) property , a terminology that was introduced in [6]for modules in F q [ x ] ℓ with respect to multiplication. For linearized polynomialsit was formulated and proven in [9]. 3 heorem 8 (PLM property,[9]) . Let M be a module in L q ( x, q m ) ℓ with minimalbasis B = { b (1) , . . . , b ( k ) } . Then for any = f ∈ M , written as f = a ( x ) ◦ b (1) + · · · + a k ( x ) ◦ b ( k ) , where a ( x ) , . . . , a k ( x ) ∈ L q ( x, q m ) , we have lm( f ) = max ≤ i ≤ k ; a i ( x ) =0 { lm( a i ) ◦ lm( b ( i ) ) } where (with slight abuse of notation) lm( a i ( x )) denotes the term of a i ( x ) ofhighest q -degree. L q ( x, q m ) ℓ We will now investigate a special family of bases, called Gr¨obner bases, forsubmodules of L q ( x, q m ) ℓ . Definition 9.
Let M ⊆ L q ( x, q m ) ℓ be a submodule. A subset B ⊂ M is calleda Gr¨obner basis of M if the leading terms of B span a left module that containsall leading terms of M .It is well-known that a Gr¨obner basis of a module M in F q [ x ] ℓ (equippedwith normal multiplication) generates M . We will now show the analog forlinearized polynomials. Theorem 10.
Let M be a module in L q ( x, q m ) ℓ with Gr¨obner basis B . Then B generates M .Proof. Let f ∈ M and B = { b (1) , . . . , b ( k ) } ⊂ L q ( x, q m ) ℓ . Since B is a Gr¨obnerbasis there exist h ( x ) , . . . , h k ( x ) ∈ L q ( x, q m ) such thatlt( f ) = k X j =1 h j (lt( b ( j ) )) . One sees that lt( f ) can only be a combination of the elements of the Gr¨obnerbasis that have the same leading position as f . Without loss of generality assumethat this is the case for b (1) , . . . , b ( k ′ ) , k ′ ≤ k . Thenlt( f ) = k ′ X j =1 h j (lt( b ( j ) )) = k ′ X j =1 h j ( b ( j ) m j x [ m j ] e lpos( f ) ) , where b ( j ) m j x [ m j ] e lpos( f ) is the leading term of b ( j ) . Denote m − := min { m j | j =1 , . . . , k ′ } . Then lt( f ) = k ′ X j =1 h j ( b ( j ) m j x [ m j − m − ] ( x [ m − ] e lpos( f ) ))= k ′ X j =1 h j ( b ( j ) m j x [ m j − m − ] ) ◦ ( x [ m − ] e lpos( f ) )4nd thus x [ m − ] e lpos( f ) symbolically divides lt( f ). Furthermore, there exists1 ≤ i ≤ k ′ such that x [ m − ] e lpos( f ) = lm( b ( i ) ).Now reduce f modulo G until it is minimal and call the resulting vector r ∈ L q ( x, q m ) ℓ . Hence there exist h ( x ) , . . . , h k ( x ) ∈ L q ( x, q m ) such that f − r = k X i =1 h i ( b ( i ) )which implies that f − r ∈ M . If r = 0, then f = P ki =1 h i ( b ( i ) ). We willnow show by contradiction that r = 0 is not possible. If r = 0 then r = f − P ki =1 h i ( b ( i ) ) ∈ M , since f ∈ M . Then, by the first part of the proof, thereexists h ( x ) ∈ L q ( x, q m ) and 1 ≤ i ≤ k such thatlt( r ) = h (lm( g i ))which means that r could be further reduced modulo G , which contradicts theminimality assumption. Thus, we have shown that any f ∈ M can be generatedby the elements of B .Thus, we have shown that any Gr¨obner basis of a module is actually a basisof this module. Clearly, the other way around is not true, i.e. not every basisis a Gr¨obner basis, but for minimal bases the reverse implication also holds, asshown in the following. Theorem 11.
Any minimal basis B of a module M ⊆ L q ( x, q m ) ℓ is a minimalGr¨obner basis of M .Proof. Let f ∈ M . Since any minimal basis of a module in L q ( x, q m ) ℓ has atmost ℓ elements, we can assume B = { b (1) , . . . , b ( ℓ ′ ) } , where ℓ ′ ≤ ℓ . There exist a ( x ) , . . . , a ℓ ′ ( x ) ∈ L q ( x, q m ) such that P i a i ( x ) ◦ b ( i ) = f . Then by Theorem 8lm( f ) = max ≤ i ≤ ℓ ′ ; a i =0 { lm( a i ) ◦ lm( b ( i ) ) } , i.e. lm( f ) and thus also lt( f ) is in the module spanned by all lm( b ( i ) ), i =1 , . . . , ℓ ′ .Finally, we show the existence of Gr¨obner bases of modules in L q ( x, q m ) ℓ . Theorem 12.
For any module M ⊆ L q ( x, q m ) ℓ there exists a finite minimalGr¨obner basis.Proof. Without restriction assume that M contains elements with leading po-sition i for all i ∈ { , . . . , ℓ } . Define f min ,i as the (non-unique) f ∈ M withlpos( f ) = i whose leading monomial is minimal, for i = 1 , . . . , ℓ . Then B = { f min , , . . . , f min ,ℓ } forms a Gr¨obner basis of M , since any leading term of M is an element of the module generated by the leading terms of B . To see this,denote an arbitrary leading term of M by c i x [ j i ] e i and lt( f min ,i ) = c m x [ j m ] e i ;then j i ≥ j m and c i x [ j i ] e i = (cid:18) c i c [ j i − j m ] m x [ j i − j m ] (cid:19) ◦ lt( f min ,i ) . Clearly, B is finite and the leading positions of all its elements are distinct.5 Conclusion
Gr¨obner bases for modules over F q [ x ] are well-known and have been extensivelystudied. In this work we have translated some of the definitions and results ofGr¨obner bases from the polynomial ring F q [ x ], equipped with multiplication,to the linearized polynomial ring L q ( x, q m ), equipped with composition. Itturns out, that, despite the different operation used in the ring of linearizedpolynomials, all results covered in this work hold in both settings. References [1] W. W. Adams and P. Loustaunau.
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