Modular Algorithms for Computing Minimal Associated Primes and Radicals of Polynomial Ideals Toru Aoyama Masayuki Noro Kobe University Rikkyo University Department of Mathematics Department of Mathematics Graduate school of Science Rikkyo University Department of Mathematics ISSAC 2018, July 18, 2018 in New York, USA
Notations R : ring K : field X = { x 1 , . . . , x n } R [ X ] : polynomial ring over R 2 / 38
Abstract Modular algorithms avoid the swell of coefficients which makes ideal computations slow-down. For computational targets in R , modular algorithms choose projection maps R to R ′ , take projected images of targets and compute in R ′ then reconstruct the real computed results in R . We apply the Chinese Remainder Theorem (CRT) for Laplagne’s algorithm. Most basically, CRT utilizes mappings: Z → F p ( p : prime number). We utilize mappings: K [ u ] → K , substituting maps for u . In order for this method to work correctly, the shape of each modular component must coincide with that of the corresponding component of the ideal. 3 / 38
Contents 1 Basic facts 2 New algorithm 3 Experimentations 4 / 38
Basic facts 5 / 38
Laplagne’s Algorithm (Laplagne. S (2006)) √ Laplagne proposes algorithms for computing minAss( I ) and I . LminAss( I ) and Lradical( I ) Int ← ⟨ 1 ⟩ , MA ← ∅ , Rad ← ⟨ 1 ⟩ √ while Int \ I ̸ = ∅ do √ choose g ∈ Int \ I J ← I : g ∞ U ← a maximal independent set of J . J ← J K ( U )[ X \ U ] { P 1 , . . . , P n } ← zeroMinAss ( J ) PJ ← { P 1 ∩ K [ X ] , . . . , P n ∩ K [ X ] } MA ← MA ∪ PJ ∩ Int ← Int ∩ P P ∈ PJ Rad ← Rad ∩ ( zeroRadical ( J ) ∩ K [ X ]) end while return MA, Rad 6 / 38
zeroMinAss zeroMinAss( I ) Input: a zero-dimensional ideal I = ⟨ f 1 , . . . , f k ⟩ ⊂ K [ X ] ( char ( K ) = 0 ) Output: minAss( I ) result ← ∅ choose a random a ∈ K n − 1 and I ′ ← φ a ( I ) ( φ a ( x i ) = x i for i < n , φ a ( x n ) = x n + ∑ n − 1 i =1 a i x i ) compute the reduced Gröbner basis G of I ′ w.r.t. < lex factorize g = g m 1 . . . g m s ∈ G ∩ K [ x n ] 1 s For i = 1 to s P ′ i ← primaryTest ( ⟨ I ′ , g i ⟩ ) If P ′ i ̸ = ⟨ 0 ⟩ P i ← φ − 1 a ( P ′ i ) result ← result ∪{ P i } Else result ← result ∪ zeroMinAss ( ⟨ I, φ − 1 a ( g i ) ⟩ ) EndIf EndFor Return result Coordinate changes are performed to make ideals in general position. zeroMinAss contains factorizations of polynomials. 7 / 38
CRT for Laplagne’s Algorithm Laplagne’s algorithm contains zeroMinAss . zeroMinAss contains factorizations of polynomials. In many cases, a factorization over F p produces more factors than over Q . Laplagne’s algorithm regards some variables U ⊂ X as parameters. We utilize mappings: K [ u ] → K ( u ∈ U ) recursively. These mappings reduce the number of parameters and keep the characteristic of coefficient fields 0 . 8 / 38
Chinese Remainder Theorem Let R be a commutative ring and I 1 , . . . , I s pairwise comaximal ideals in R . For r 1 , . . . , r s ∈ R , there exists y ∈ R satisfying y ≡ r 1 mod I 1 . . . y ≡ r s mod I s . y is unique modulo ∩ s i =1 I i . CRT can be applied in two typical situations: R = Z or R = K [ u ] . 9 / 38
Lagrange’s Interpolation in Z Let p 1 , . . . , p s be distinct prime numbers from each other, p = p 1 · · · p s and I 1 = ⟨ p 1 ⟩ , . . . , I s = ⟨ p s ⟩ . Then for 1 ≤ i ≤ s , a i , b i ∈ Z such that a i ( p/p i ) + b i p i = 1 can be computed by the extended Euclidean algorithm. For any r 1 , . . . , r s ∈ Z , the unique y satisfying conditions in CRT is given by y = r 1 L 1 + · · · + r s L s ( where L i = a i ( p/p i )) . 10 / 38
Lagrange’s Interpolation in K [ u ] Let k 1 , . . . , k s ∈ K be distinct elements from each other, I 1 = ⟨ u − k 1 ⟩ , . . . , I s = ⟨ u − k s ⟩ and ( u − k 1 ) · · · ( u − k i − 1 )( u − k i +1 ) · · · ( u − k s ) L i = ( k i − k 1 ) · · · ( k i − k i − 1 )( k i − k i +1 ) · · · ( k i − k s ) . Then the unique y satisfying conditions in CRT is given by y = r 1 L 1 + · · · + r s L s . 11 / 38
CRT Let r 1 , r 2 ∈ K [ u ] , I 1 , I 2 comaximal ideals ∈ K [ u ] . We name the interpolation r 1 (mod I 1 ) and r 2 (mod I 2 ) CRT ( r 1 , r 2 , I 1 , I 2 ) . α d α x α ∈ K [ u ][ X ] , we define α c α x α , g = ∑ For f = ∑ α CRT ( c α , d α , I 1 , I 2 ) x α . CRT ( f, g, I 1 , I 2 ) = ∑ For F = { f 1 , . . . , f s } , G = { g 1 , . . . , g s } ⊂ K [ u ][ X ] where LM ( f i ) ’s and LM ( g i ) ’s are distinct respectively and LM ( f i ) = LM ( g i ) , we define CRT ( F, G, I 1 , I 2 ) = { CRT ( f i , g i , I 1 , I 2 ) | 1 ≤ i ≤ s } . For F = { F 1 , . . . , F t } and G = { G 1 , . . . , G t } where CRT ( F i , G i , I 1 , I 2 )’s are defined, we define CRT ( F , G , I 1 , I 2 ) = { CRT ( F i , G i , I 1 , I 2 ) | 1 ≤ i ≤ t } . 12 / 38
Rational function reconstruction Our target is the reduced Gröbner basis G of a minimal associated prime of an ideal I over K ( u ) . If we apply CRT for the modular images computed over K , what we obtain is an object G ′ over K [ u ] . If a coefficient c ( u ) appearing in G is not a polynomial we have to recover c ( u ) from the corresponding polynomial coefficient in G ′ . Theorem (Gathen-Gerhard (2003)) Let f, M ∈ K [ x ] , deg( f ) < deg( M ) = n > 0 and r i , s i , t i ∈ K [ x ] be the j -th row in extended Euclidean Algorithm for M, f ,where j is minimal such that deg( r j ) < k . There exist polynomials r, t ∈ K [ x ] satisfying r ≡ tf (mod M ) , deg( r ) < k, deg( t ) ≤ n − k, namely r = r j , t = t j . If in addition gcd( r j , t j ) = 1 , then r, t also satisfy gcd( t, M ) = 1 , rt − 1 ≡ f (mod M ) , deg( r ) < k, deg( t ) ≤ n − k 13 / 38
Algorithm for rational function reconstruction RFR ( f, M ) Input: polynomials f, M ∈ K [ x ] Output: g, h ∈ K [ x ] s.t. f ≡ g/h (mod M ) , h is monic and gcd( g, h ) = 1 r 0 ← M , r 1 ← f t 0 ← 0 , t 1 ← 1 i ← 1 While 2 deg( r i ) > deg( M ) R i ← NF r i − 1 , { r i } Q ← ( r i − 1 − R i ) /r i r i +1 ← R i , t i +1 ← t i − 1 − Qt i i ← i + 1 EndWhile Return ( r i , t i ) 14 / 38
RFR We also utilize the algorithm RFR for reconstructing coefficients of polynomials, ideals and a set of ideals. Let ⟨ M ⟩ = ∩ i ⟨ u − k i ⟩ ( k i ∈ K ) ∈ K [ u ] . α c α x α ∈ K [ u ][ X ] , we denote RFR ( f, M ) For f = ∑ α RFR ( c α , M ) x α . = ∑ For F ⊂ K [ u ][ X ] , we define RFR ( F, M ) = { RFR ( f, M ) | f ∈ F } . For F = { F 1 , . . . , F s } where RFR ( F i , M )’s are defined, we define RFR ( F , M ) = { RFR ( F, M ) | F ∈ F } . 15 / 38
Remark When we reconstruct g ( u ) h ( u ) ∈ K ( u ) ( gcd( g, h ) = 1 ) from f ( u ) ∈ K [ u ] by RFR , we need more than deg( g ) + deg( h ) ideals ⟨ u − k i ⟩ ( k i ∈ K and h ( k i ) ̸ = 0 ). We say that the output of RFR is stable if we have more than deg( g ) + deg( h ) ideals. We say that the output is pseudo stable if RFR ( f ( u ) , M ) = RFR ( f ( u ) , M ′ ) ,where M = ∩ r i =1 ⟨ u − k i ⟩ , M ′ = ∩ s i =1 ⟨ u − k i ⟩ ( r < s ). We regard the pseudo stable output as a candidate of the unique rational function. 16 / 38
New Algorithm 17 / 38
Luckiness (extensions of Noro-Yokoyama (2016)) Let u / ∈ X be a variable, F a subset of K ( u )[ X ] , G the reduced Gröbner basis of ⟨ F ⟩ and k ∈ K , then ⟨ u − k ⟩ is a prime ideal in K [ u ] . 1 K [ u ] ( u − k ) := { f g | f, g ∈ K [ u ] , g ( k ) ̸ = 0 } . 2 ϕ ( u − k ) : K ( u ) → K ; f �→ f ( k ) . We denote projection maps K [ u ] ( u − k ) → K and K [ u ] ( u − k ) [ X ] → K [ X ] by the same symbol α c α x α �→ ∑ α ϕ ( u − k ) ( c α ) x α ( c α is g �→ f ( k ) ϕ ( u − k ) such that f g ( k ) and ∑ the coefficient of c α x α ). 3 I ( u − k ) ( F ) := ⟨ ϕ ( u − k ) ( f ) | f ∈ F ⟩ . 4 ⟨ u − k ⟩ is said to be weak permissible for F if F ⊂ K [ u ] ( u − k ) . ⟨ u − k ⟩ is said to be permissible for F if ⟨ u − k ⟩ is weak permissible for F and ϕ ( u − k ) ( LC ( f )) ̸ = 0 for all f ∈ F 5 Let √ ⟨ G ⟩ = ∩ m i =1 P i be the prime decomposition and G i the reduced Gröbner basis of P i . ⟨ u − k ⟩ is said to be effectively minass lucky for G if ⟨ u − k ⟩ is permissible for G and G i ( i = 1 , . . . , m ), √ I ( u − k ) ( G ) = ∩ m i =1 Q i is the prime decomposition and ϕ ( u − k ) ( G i ) is the reduced Gröbner basis of Q i . 18 / 38
Fundamental lemmas Lemma Let G be a Gröbner basis (respectively the reduced Gröbner basis) of I ⊂ K ( u )[ X ] ( u / ∈ X ). If an ideal ⟨ u − k ⟩ is permissible for G , then ϕ ( u − k ) ( G ) is a Gröbner basis (respectively the reduced Gröbner basis) of I ( u − k ) ( G ) . Lemma ∈ X be a parameter, I ⊂ K ( u )[ X ] an ideal and G = { g 1 , . . . , g m } Let u / the reduced Gröbner basis of I . If k ∈ K , ⟨ u − k ⟩ is permissible for G and I ( u − k ) ( G ) is a prime ideal in K [ X ] , then I is a prime ideal in K ( u )[ X ] . Lemma Let P, Q be ideals in K ( u )[ X ] , G = { g 1 , . . . , g s } the reduced Gröbner basis of P , H = { h 1 , . . . , h r } the reduced Gröbner basis of Q . If k ∈ K and ⟨ u − k ⟩ is permissible for G, H and ⟨ ϕ ( u − k ) ( G ) ⟩ ̸⊂ ⟨ ϕ ( u − k ) ( H ) ⟩ , then P ̸⊂ Q . 19 / 38
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