Additive Decomposition of Polynomials over Unique Factorization - PowerPoint PPT Presentation

Additive Decomposition of Polynomials over Unique Factorization Domains Manar Benoumhani Supervisor: Dr.Leila Benferhat Department of Mathematics University of sciences and technology Houari Boumediene October 21, 2019 Manar Benoumhani (Department of Mathematics University of sciences and technology Houari Boumediene ) USTHB October 21, 2019 1 / 30

Outline 1 Preliminaries. 2 The Diamond Product over F q . 3 Additive Decompositon Over UFD’s. Manar Benoumhani (Department of Mathematics University of sciences and technology Houari Boumediene ) USTHB October 21, 2019 2 / 30

Preliminaries • F q : The finite field of order q where q = p s , p is prime. • R [ x ]: The ring of polynomials with coefficients in R . Definition Let a , b and c be elements of an integral domain R . 1 a and b are associates, a = ub , where u is a unit of R . 2 If a is not zero, a is called an irreducible if it is not a unit and, whenever a = bc , then b or c is a unit. 3 If a is not zero, a is called a prime if a is not a unit and a | bc implies a | b or a | c . Manar Benoumhani (Department of Mathematics University of sciences and technology Houari Boumediene ) USTHB October 21, 2019 3 / 30

Preliminaries Definition (UFD) An integral domain R is a unique factorization domain if 1 Every nonzero element of R that is not a unit can be written as a product of irreducibles of R ; and 2 The factorization into irreducibles is unique up to associates and the order in which the factors appear. Manar Benoumhani (Department of Mathematics University of sciences and technology Houari Boumediene ) USTHB October 21, 2019 4 / 30

Preliminaries Definition (UFD) An integral domain R is a unique factorization domain if 1 Every nonzero element of R that is not a unit can be written as a product of irreducibles of R ; and 2 The factorization into irreducibles is unique up to associates and the order in which the factors appear. Theorem • Let F be a field. Then, F [ x ] is a UFD. • If R is a UFD, then R [ x ] is a UFD. Manar Benoumhani (Department of Mathematics University of sciences and technology Houari Boumediene ) USTHB October 21, 2019 4 / 30

Preliminaries Resultant n m a i x i and g ( x ) = b i x i be two polynomials over a commutative ring R with � � Let f ( x ) = i =0 i =0 identity. The Sylvester matrix of f and g is the following ( n + m ) × ( n + m ) matrix:   a m · · · a 0 ... ...   · · ·     a m · · · a 0   Sylv =   b n · · · b 0     ... ...   · · ·   b n · · · b 0 Definition (Resultant) The resultant of two polynomials f and g is defined by: Res x ( f , g ) = det( Sylv ) Manar Benoumhani (Department of Mathematics University of sciences and technology Houari Boumediene ) USTHB October 21, 2019 5 / 30

Preliminaries Properties of Resultants Theorem n m Let f ( x ) = a n � ( x − α i ) and g ( x ) = b m � ( x − β j ) be two polynomials of an integral domain i =1 j =1 R with indeterminates α 1 , . . . , α n and β 1 , . . . , β m .Then m � Res x ( f , g ) = ( − 1) nm b n f ( β i ) . (1) m i =1 n Res x ( f , g ) = a m � g ( α i ) . (2) n i =1 n m Res x ( f , g ) = a m n b n � � ( α i − β j ) (3) m i =1 j =1 Manar Benoumhani (Department of Mathematics University of sciences and technology Houari Boumediene ) USTHB October 21, 2019 6 / 30

Preliminaries Theorem (R¨ udiger G.K. Loos 1973) n m Let f ( x ) = a n � ( x − α i ) and g ( x ) = b m � ( x − β j ) be two polynomials of positive degree i =1 j =1 over an integral domain R with roots α 1 , . . . , α n and β 1 , . . . , β m respectively. Then the polynomial n m � � r ( x ) = ( − 1) nm ga m n b n ( x − γ ij ) m i =1 j =1 has nm roots, not necessarily distinct, suct that: 1 r ( x ) = Res y ( f ( x − y ) , g ( y )) , γ ij = α i + β j , g = 1 . 2 r ( x ) = Res y ( f ( x + y ) , g ( y )) , γ ij = α i − β j , g = 1 . 3 r ( x ) = Res y ( y m f ( x / y ) , g ( y )) , γ ij = α i β j , g = 1 . 4 B − m r ( x ) = Res y ( f ( xy ) , g ( y )) , γ ij = α i /β j , g = ( − 1) nm g (0) n � b n m , g (0) � = 0 . 0 Manar Benoumhani (Department of Mathematics University of sciences and technology Houari Boumediene ) USTHB October 21, 2019 7 / 30

Preliminaries Proof. The proof is based on (1)in all cases. Manar Benoumhani (Department of Mathematics University of sciences and technology Houari Boumediene ) USTHB October 21, 2019 8 / 30

Preliminaries Proof. The proof is based on (1)in all cases. Corollary Except for [4], the polynomial r ( x ) is monic if f and g are. Manar Benoumhani (Department of Mathematics University of sciences and technology Houari Boumediene ) USTHB October 21, 2019 8 / 30

Part I Additive decomposition for polynomials over F q Manar Benoumhani (Department of Mathematics University of sciences and technology Houari Boumediene ) USTHB October 21, 2019 9 / 30

The diamond product • Let Ω be the algebraic closure of F q and ∅ � = G ⊂ Ω such that ∀ α ∈ G , σ ( α ) ∈ G where σ is the Frobenius automorphism of Ω. Manar Benoumhani (Department of Mathematics University of sciences and technology Houari Boumediene ) USTHB October 21, 2019 10 / 30

The diamond product • Let Ω be the algebraic closure of F q and ∅ � = G ⊂ Ω such that ∀ α ∈ G , σ ( α ) ∈ G where σ is the Frobenius automorphism of Ω. • There is defined a binary operation ⋄ on G such that: ∀ α, β ∈ G : σ ( α ⋄ β ) = σ ( α ) ⋄ σ ( β ) . Manar Benoumhani (Department of Mathematics University of sciences and technology Houari Boumediene ) USTHB October 21, 2019 10 / 30

The diamond product • Let Ω be the algebraic closure of F q and ∅ � = G ⊂ Ω such that ∀ α ∈ G , σ ( α ) ∈ G where σ is the Frobenius automorphism of Ω. • There is defined a binary operation ⋄ on G such that: ∀ α, β ∈ G : σ ( α ⋄ β ) = σ ( α ) ⋄ σ ( β ) . • M G [ q , x ] denote the set of all monic polynomials f in F q such that: 1 The degree of f ≥ 1. 2 All the roots of f lie in G . Manar Benoumhani (Department of Mathematics University of sciences and technology Houari Boumediene ) USTHB October 21, 2019 10 / 30

The diamond product • Let f , g ∈ M G [ q , x ] such that f = � ( x − α ) and g = � ( x − β ), then: α β Manar Benoumhani (Department of Mathematics University of sciences and technology Houari Boumediene ) USTHB October 21, 2019 11 / 30

The diamond product • Let f , g ∈ M G [ q , x ] such that f = � ( x − α ) and g = � ( x − β ), then: α β Definition The diamond product of f and g is defined as: � � f ⋄ g = ( x − α ⋄ β ) (4) α β Manar Benoumhani (Department of Mathematics University of sciences and technology Houari Boumediene ) USTHB October 21, 2019 11 / 30

The diamond product • Let f , g ∈ M G [ q , x ] such that f = � ( x − α ) and g = � ( x − β ), then: α β Definition The diamond product of f and g is defined as: � � f ⋄ g = ( x − α ⋄ β ) (4) α β • Clearly, if deg( f ) = n and deg( g ) = m then deg( f ⋄ g ) = nm . Manar Benoumhani (Department of Mathematics University of sciences and technology Houari Boumediene ) USTHB October 21, 2019 11 / 30

The diamond product Example 1 Let G = Ω and α ⋄ β = α + β . We’ll have � � f ⋄ g = ( x − ( α + β )) (5) α β � � = g ( x − α ) = f ( x − β ) , (6) α β = f ∗ g . (7) 2 If G = Ω / { 0 } and α ⋄ β = αβ , then: � � f ⋄ g = ( x − αβ ) , (8) α β � � α m g ( x /α ) = β n f ( x /β ) , = (9) α β = f ◦ g . (10) Manar Benoumhani (Department of Mathematics University of sciences and technology Houari Boumediene ) USTHB October 21, 2019 12 / 30

The diamond product Example Let f = x 2 + x + 1 and g = x 3 + x + 1 be two polynomials in F 2 [ x ]. In Ω[ x ], we have f = ( x − α )( x − α 2 ) , g = ( x − β )( x − β 2 )( x − β 4 ) where α and β are the roots of f and g respectively. Applying (6) and (8) , it follows that: f ∗ g = g ( x − α ) g ( x − α 2 ) , = x 6 + x 5 + x 3 + x 2 + 1 . f ◦ g = α 3 g ( x /α ) α 6 g ( x /α 2 ) = ( x 3 + α 2 x + α 3 )( x 3 + α 4 x + α 6 ) , = x 6 + x 4 + x 2 + x + 1 . f ∗ f = x 2 ( x + 1) 2 and f ◦ f = ( x + 1) 2 ( x 2 + x + 1). Manar Benoumhani (Department of Mathematics University of sciences and technology Houari Boumediene ) USTHB October 21, 2019 13 / 30

The diamond product Theorem The diamond product is a binary operation on M G [ q , x ] . • The units of M G [ q , x ] are the polynomials x − c where c is a unit in G . • f and g are associates ( f ∼ g ) iff f = ( x − c ) ⋄ g for some unit x − c . • A polynomial h in M G [ q , x ] which is not a unit is said to be decomposable with respect to ⋄ iff there are polynomials f and g such that h = f ⋄ g , otherwise, h is idecomposable . Manar Benoumhani (Department of Mathematics University of sciences and technology Houari Boumediene ) USTHB October 21, 2019 14 / 30