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
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