Factoring Polynomials over Local Fields II Sebastian Pauli - PowerPoint PPT Presentation

Factoring Polynomials over Local Fields II Sebastian Pauli Department of Mathematics and Statistics University of North Carolina at Greensboro Sebastian Pauli (UNC Greensboro) Factoring Polynomials over Local Fields II 1 / 20

Polynomial Factorization and Related Algorithms Round 4 maximal order algorithm [Ford, Zassenhaus (1976)] Montes Algorithm for ideal decomposition [Montes (1999)] Polynomial Factorization [Cantor, Gordon (2000)] N 4+ ε ν ( disc Φ) 2+ ε � � O Polynomial Factorization [Ford, P., Roblot (2002)] Polynomial Factorization [P. (2001)] Montes Algorithm revisited [Guardia, Montes, Nart (2008–)] Complexity of Montes Algorithm [Ford, Veres (2010)] O ( N 3+ ε ν ( disc Φ) + N 2+ ε ν ( disc Φ) 2+ ε ) Sebastian Pauli (UNC Greensboro) Factoring Polynomials over Local Fields II 2 / 20

Notation field complete with respect to a non-archimedian valuation K O K valuation ring of K π uniformizing element in O K ν exponential valuation normalized such that ν ( π ) = 1 K residue class field O K / ( π ) of K with char K = p Φ( x ) ∈ O K [ x ] separable, squarefree, monic: the polynomial to be factored ϕ ( x ) ∈ O K [ x ] monic: an approximation to an irreducible factor of Φ( x ) Sebastian Pauli (UNC Greensboro) Factoring Polynomials over Local Fields II 3 / 20

Reducibility – Classical Hensel’s Lemma A factorization of Φ( x ) into coprime factors over the residue class field K can be lifted to a factorization of Φ( x ) over O K . Sebastian Pauli (UNC Greensboro) Factoring Polynomials over Local Fields II 4 / 20

Reducibility – Classical Hensel’s Lemma A factorization of Φ( x ) into coprime factors over the residue class field K can be lifted to a factorization of Φ( x ) over O K . Newton Polygons Each distinct segment of the Newton Polygon of Φ( x ) corresponds to a distinct factor of Φ( x ). Sebastian Pauli (UNC Greensboro) Factoring Polynomials over Local Fields II 4 / 20

Reducibility Let Φ( x ) := � N i =1 ( x − α i ) ∈ O K [ x ] and ϑ ( x ) ∈ K [ x ], then we set N � χ ϑ ( y ) := ( y − ϑ ( α i )) = res x (Φ( x ) , y − ϑ ( x )) . i =1 Sebastian Pauli (UNC Greensboro) Factoring Polynomials over Local Fields II 5 / 20

Reducibility Let Φ( x ) := � N i =1 ( x − α i ) ∈ O K [ x ] and ϑ ( x ) ∈ K [ x ], then we set N � χ ϑ ( y ) := ( y − ϑ ( α i )) = res x (Φ( x ) , y − ϑ ( x )) . i =1 Hensel Test If χ ϑ ( y ) ∈ O K [ y ] and χ ϑ ( y ) ≡ ρ ( y ) r mod ( π ) with ρ ( y ) irreducible in K we say ϑ ( x ) passes the Hensel test . Sebastian Pauli (UNC Greensboro) Factoring Polynomials over Local Fields II 5 / 20

Reducibility Let Φ( x ) := � N i =1 ( x − α i ) ∈ O K [ x ] and ϑ ( x ) ∈ K [ x ], then we set N � χ ϑ ( y ) := ( y − ϑ ( α i )) = res x (Φ( x ) , y − ϑ ( x )) . i =1 Hensel Test If χ ϑ ( y ) ∈ O K [ y ] and χ ϑ ( y ) ≡ ρ ( y ) r mod ( π ) with ρ ( y ) irreducible in K we say ϑ ( x ) passes the Hensel test . If ϑ ( x ) fails the Hensel Test we can derive a proper factorization of Φ( x ). Sebastian Pauli (UNC Greensboro) Factoring Polynomials over Local Fields II 5 / 20

Reducibility Let Φ( x ) := � N i =1 ( x − α i ) ∈ O K [ x ] and ϑ ( x ) ∈ K [ x ], then we set N � χ ϑ ( y ) := ( y − ϑ ( α i )) = res x (Φ( x ) , y − ϑ ( x )) . i =1 Hensel Test If χ ϑ ( y ) ∈ O K [ y ] and χ ϑ ( y ) ≡ ρ ( y ) r mod ( π ) with ρ ( y ) irreducible in K we say ϑ ( x ) passes the Hensel test . If ϑ ( x ) fails the Hensel Test we can derive a proper factorization of Φ( x ). Newton Test We set v ∗ Φ ( ϕ ) := min Φ( α )=0 ν ( ϕ ( α )) and say the polynomial ϕ ( x ) passes the Newton test if ν ( ϕ ( α )) = v ∗ Φ ( ϕ ) for all roots α of Φ( x ). Sebastian Pauli (UNC Greensboro) Factoring Polynomials over Local Fields II 5 / 20

Reducibility Let Φ( x ) := � N i =1 ( x − α i ) ∈ O K [ x ] and ϑ ( x ) ∈ K [ x ], then we set N � χ ϑ ( y ) := ( y − ϑ ( α i )) = res x (Φ( x ) , y − ϑ ( x )) . i =1 Hensel Test If χ ϑ ( y ) ∈ O K [ y ] and χ ϑ ( y ) ≡ ρ ( y ) r mod ( π ) with ρ ( y ) irreducible in K we say ϑ ( x ) passes the Hensel test . If ϑ ( x ) fails the Hensel Test we can derive a proper factorization of Φ( x ). Newton Test We set v ∗ Φ ( ϕ ) := min Φ( α )=0 ν ( ϕ ( α )) and say the polynomial ϕ ( x ) passes the Newton test if ν ( ϕ ( α )) = v ∗ Φ ( ϕ ) for all roots α of Φ( x ). If ϕ ( x ) fails the Newton Test we can derive a proper factorization of Φ( x ). Sebastian Pauli (UNC Greensboro) Factoring Polynomials over Local Fields II 5 / 20

Irreducibility – Certificates Let Φ( x ) ∈ O K [ x ] and ϕ ( x ) ∈ K [ x ] with χ ϕ ( y ) ∈ O K [ y ]. If ϕ ( x ) passes the Hensel test, that is, χ ϕ ( y ) = ρ ( y ) r for some irreducible ρ ( y ) ∈ K [ y ], we set F ϕ := deg ρ. If ϕ ( x ) passes the Newton test, let E ϕ be the denominator of v ∗ Φ ( ϕ ) in lowest terms. Sebastian Pauli (UNC Greensboro) Factoring Polynomials over Local Fields II 6 / 20

Irreducibility – Certificates Let Φ( x ) ∈ O K [ x ] and ϕ ( x ) ∈ K [ x ] with χ ϕ ( y ) ∈ O K [ y ]. If ϕ ( x ) passes the Hensel test, that is, χ ϕ ( y ) = ρ ( y ) r for some irreducible ρ ( y ) ∈ K [ y ], we set F ϕ := deg ρ. If ϕ ( x ) passes the Newton test, let E ϕ be the denominator of v ∗ Φ ( ϕ ) in lowest terms. Two Element Certificates A two-element certificate for Φ( x ) is a pair (Γ( x ) , Π( x )) ∈ K [ x ] 2 such that χ Γ ( t ) ∈ O K [ t ], χ Π ( t ) ∈ O K [ t ], and F Γ E Π = deg Φ. Sebastian Pauli (UNC Greensboro) Factoring Polynomials over Local Fields II 6 / 20

Irreducibility – Certificates Let Φ( x ) ∈ O K [ x ] and ϕ ( x ) ∈ K [ x ] with χ ϕ ( y ) ∈ O K [ y ]. If ϕ ( x ) passes the Hensel test, that is, χ ϕ ( y ) = ρ ( y ) r for some irreducible ρ ( y ) ∈ K [ y ], we set F ϕ := deg ρ. If ϕ ( x ) passes the Newton test, let E ϕ be the denominator of v ∗ Φ ( ϕ ) in lowest terms. Two Element Certificates A two-element certificate for Φ( x ) is a pair (Γ( x ) , Π( x )) ∈ K [ x ] 2 such that χ Γ ( t ) ∈ O K [ t ], χ Π ( t ) ∈ O K [ t ], and F Γ E Π = deg Φ. If a two-element certificate exists for Φ( x ) then Φ( x ) is irreducible. Sebastian Pauli (UNC Greensboro) Factoring Polynomials over Local Fields II 6 / 20

Termination We construct a sequence ϕ 1 ( x ) , ϕ 2 ( x ) , . . . of approximations to a factor of Φ( x ) such that ν ( ϕ 1 ( α )) < ν ( ϕ 2 ( α )) < . . . for all roots α of Φ( x ) until we find one of the situations described above. Sebastian Pauli (UNC Greensboro) Factoring Polynomials over Local Fields II 7 / 20

Termination We construct a sequence ϕ 1 ( x ) , ϕ 2 ( x ) , . . . of approximations to a factor of Φ( x ) such that ν ( ϕ 1 ( α )) < ν ( ϕ 2 ( α )) < . . . for all roots α of Φ( x ) until we find one of the situations described above. Theorem (P. 2001) If Φ( x ) ∈ O K [ x ] separable, squarefree, monic, – ϕ ( x ) ∈ O K [ x ] monic, – ν ( ϕ ( α )) > 2 · ν ( disc Φ) / deg Φ for all roots α of Φ( x ), and – the degree of any irreducible factor of Φ( x ) is greater than or equal to deg ϕ , then deg ϕ = deg Φ and Φ( x ) is irreducible over K . Sebastian Pauli (UNC Greensboro) Factoring Polynomials over Local Fields II 7 / 20

Sketch of an Algorithm Input: a monic, separable, squarefree polynomial Φ( x ) ∈ O K [ x ] Output: a proper factorization of Φ( x ) or a two-element certificate for the irreducibility of Φ( x ) t ← 1, ϕ 1 ← x , E ← 1, F ← 1. Repeat: If ϕ t ( x ) fails the Newton test: return a factorization of Φ( x ). 1 If we find more ramification: increase E . 2 . . . 3 If we find more inertia: increase F . 4 . . . 5 If E · F = deg Φ: return a two-element certificate. 6 Find ϕ t +1 ( x ) ∈ O K [ x ] with v ∗ Φ ( ϕ t +1 ) > v ∗ Φ ( ϕ t ), deg ϕ t +1 = EF . 7 t ← t + 1 8 Sebastian Pauli (UNC Greensboro) Factoring Polynomials over Local Fields II 8 / 20

Main Steps Newton Test Round 4: Newton Polygon of the Characteristic Polynomial χ ϕ ( y ) of ϕ ( x ) Montes: ϕ -adic Expansion of Φ( x ) Hensel Test Round 4: Characteristic Polynomial χ ϕ e ψ − 1 ( y ) of ϕ e ( x ) ψ − 1 ( x ) where Φ ( ϕ e ) v ∗ Φ ( ψ ) = v ∗ Montes: Residual Polynomial Construction of Next ϕ Sebastian Pauli (UNC Greensboro) Factoring Polynomials over Local Fields II 9 / 20

The 1st Iteration – Newton Polygon ϕ 1 ( x ) = x If the Newton polygon of Φ( x ) consists of more than one segment, then we can derive a factorization of Φ( x ). Otherwise let − h 1 E 1 be the slope of the Newton polygon in lowest terms. Φ ( x ) = h 1 Then ν ( α ) = v ∗ E 1 for all roots α of Φ( x ). E 1 is a divisor of the ramification indices of all K ( α ) where α is a root of Φ( x ). Sebastian Pauli (UNC Greensboro) Factoring Polynomials over Local Fields II 10 / 20

The 1st Iteration – Residual Polynomial ν (Φ i ) Newton Polygon of Φ( x ) = x N + � N − 1 ✻ i =0 Φ i x i ν (Φ 0 ) r r ❜❜❜❜❜❜❜❜❜❜❜❜❜ Φ ( x ) = − ν (Φ 0 ) r with slope − h 1 E 1 = − v ∗ where gcd( h 1 , E 1 ) = 1 N r r r ✲ i ❜ 0 r N Sebastian Pauli (UNC Greensboro) Factoring Polynomials over Local Fields II 11 / 20