An Efficient Implementation for Computing Gr obner bases over - - PowerPoint PPT Presentation

An Efficient Implementation for Computing Gr¨

• bner bases over

algebraic number fields

Masayuki Noro Kobe University

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 1

Multiple algebraic extension

An algebraic number field can be represented as K = Q(α1, . . . , αl) = R/J R = Q[t1, . . . , tl], J = D D = {m1(t1), m2(t1, t2), . . . , ml(t1, . . . , tl)} J : the defining ideal of K – zero-dimensional maximal ideal mi(α1, . . . , αi−1, ti) : irreducible over Ki−1 = Q(α1, . . . , αi−1)

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 2

Gröbner basis computation over K

Direct computation over K Field operations are hard Remainder computation by D is necessary for each field operations. Coefficient growth It is more serious than over Q.

• ver Q

Removal of integer contents Shortcut by modular computation

• ver K

How can we avoid the coefficient growth?

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 3

An alternative method over Q

S = K[x1, . . . , xn], T = Q[x1, . . . , xn, t1, . . . , tl] ˜ B = {g1(x, t), . . . , gd(x, t)} ⊂ T B = {g1(x, α), . . . , gd(x, α)} ⊂ S ≺ : a term order inS ≺K : a product order of ≺ and the lex order in R Theorem Let ˜ G be the reduced Gröbner basis of ˜ I = ˜ B ∪ D with respect to ≺K. Then ( ˜ G \ D)|t=α is the reduced Gröbner basis of I = B with respect to ≺.

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 4

An observation

Computation under ≺K Many bases of the form tbxa + lower are produced before a monic element xa + lower is produced. ⇒ tbxa + lower are all redundant Reason Let HCR(f) be the head coefficient of f w.r.t. ≺ ⇒ The sequence {ra} (ra = tbxa + lower) corresponds to the computation of HCR(r)−1 by Euclid algorithm for some normal form r.

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 5

Possible difficulties

If the extension degree is high.. Increase of the number of S-pairs It simply increases the total cost. It may introduce some bad effect in the selection strategy. Inefficiency of the computation of HCR(h)−1 Inverse computation of an algebraic number by non-modular Euclid algorithm ⇒ usually slower than modular computation

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 6

A simple idea

• 1. Apply the alternative method over Q for each

normal form computation Removal of integer contents is applicable. Modular methods (trace algorithm etc.) are applicable.

• 2. The normal form is made monic before added to

the basis set tbxa + lower ∈ I ⇒ xa + lower ∈ I Intermediate elements are not produced. ⇒ It simulates the computation over K.

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 7

Outline of the algorithm

L ← {{f, g} | f, g ∈ ˜ B, f = g}; G ← ˜ B while L = ∅ do {f, g} ← an element of L; L ← L \ {{f, g}} ˜ r(x, t) ← NFD∪G(S(f, g)) if ˜ r = 0 then u(t) ← the inverse of HCR(˜ r) mod J r ← NFD(u˜ r) L ← L ∪ {{f, r} | f ∈ G};G ← G ∪ {r} end if end while return G

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 8

Operations to be considered

• 1. Simplification of algebraic numbers

If K is a multiple extension, the simplification is not trivial.

• 2. Computation of an inverse in K

Modular computations should be applied.

• 3. Configuration of Buchberger algorithm

We have to be careful for avoiding coefficient growth. ⇒ homogenized trace algorithm

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 9

• 1. Simplification of algebraic numbers

Efficiency depends on the order (priority) of defining polynomials to be used for simplification. An example m1(t1) = t20

1 + t19 1 + 2, m1(α1) = 0

m2(t1, t2) = t30

2 + (t19 1 + t18 1 + 1)t29 2 + 1, m2(α1, α2) = 0

Simplification of α58

2

Use mk with the smallest possible k : 0.02sec Use mk with the largest possible k : 2sec

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 10

A simplification algorithm

r ← 0 while f = 0 do if HT(f) is reduced with respect to D then r ← r + HT(f) f ← f − HT(f) else k ← the smallest k such that HT(mk)|HT(f) (∗) f ← f − HC(f) ·

HT(f) HT(mk)mk

endif end while return r

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 11

A proposition on a degree bound

Proposition Let f1 be the right hand side of (∗) in the simplification algorithm. Then we have degti(f1) ≤    MAX(2(di − 1), degti(f)) (i ≤ k − 1) degti(f) (i ≥ k) (di = degti(mi)) If an input f(t) is reduced w.r.t. D ⇒ degti(f) ≤ 2(di − 1) holds during the execution of the algorithm.

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 12

• 2. Computation of an inverse in K

Let M = {s1, . . . , sd} be the set of monomials which spans R/D over Q, where d = dimQ R/D. Algorithm for computing f(α)−1 Convert

i ciNFG(fsi) = 1 (⇔ f( i cisi) = 1)

into a system of linear equations Ac = b with respect to c = (c1, . . . , cd)T. a = (a1, . . . , ad)T ← the solution of Ac = b (Apply a modular method such as Hensel lifting or CRT.) return

i aisi

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 13

• 3. Homogenized trace algorithm

Bh ← the homogenization of B loop p ← a new prime Gh ← Candidate(Bh, p) s.t. Gh ⊂ Bh if Gh = failure then G ← the dehomogenization of Gh G ← {g ∈ G | HT(h) |HT(g) for all h ∈ G \ {g}} if G is a Gröbner basis of G and NFG(f) = 0 for all f ∈ B then return G endif end loop

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 14

Realization of Candidate

Buchberger algorithm+modular trace

L ← {{f, g} | f, g ∈ B, f = g}; G ← B; Gp ← B mod p while L = ∅ do {f, g} ← an element of L; L ← L \ {{f, g}} if NFGp(S(f mod p, g mod p)) = 0 then r ← NFG(S(f, g)) u(t) ← the inverse of HCR(r) mod J; r ← NFD(ur) if HC(r) mod p = 0 then return failure L ← L ∪ {{f, r} | f ∈ G} G ← G ∪ {r}; Gp ← Gp ∪ {r mod p} end if end while return G

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 15

An improvement : frequent inter-reduction

Inter-reduction : a natural operation for a homogeneous input Useful to reduce the coefficient sizes over Q ⇒ also useful over an algebraic number field Old implementation in Risa/Asir Inter-reduction is done only after processing all the S-polys of the same total degree. Current implementation Inter-reduction is executed every after k new basis elements are generated (k = 6 by default.)

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 16

Related functions

newalg(DefPoly) It generates a root of DefPoly af(Poly,AlgList) It factors Poly a root of Poly over Q(AlgList). nd gr trace(PL,V L,Homo,Trace,Ord) A general function to compute Gröbner bases. If Homo = Trace = 1, it executes the homogenized trace algorithm with Candidate=Buchberger+trace.

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 17

Experiments in Risa/Asir

Purposes Compare the well-known algorithm over Q and the new one. Compare the non-trace and the trace algorithm Non-trace one computes the GB of the homogenized inputs. Examine the effect of inter-reduction Machine : Linux/Xeon 3.4GHz, 6GB of memory

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 18

Example 1 : C7|c7=ω

C7|c7=ω; ω is a root of m(c7), where m(c7) is an irreducible factor of the minimal polynomial of c7 in Q[c1, . . . , c7]/C7.

• 1. m(c7) = c2

7 + 5c7 + 1

• 2. m(c7) = c6

7 + c5 7 + c4 7 + c3 7 + c2 7 + c7 + 1

• 3. m(c7) = c12

7 −5c11 7 +24c10 7 −115c9 7+551c8 7−2640c7 7+

12649c6

7 − 2640c5 7 + 551c4 7 − 115c3 7 + 24c2 7 − 5c7 + 1

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 19

Timings for example 1

non-trace trace #basis check monic

• ld trace

1/Q(ω) 462 198 306 83 1480 1/Q 876 119 675 – – 2/Q(ω) 22 9.3 268 0.2 1.4 13 2/Q 262 74 588 0.2 – – 3/Q(ω) 544 256 306 128 1810 3/Q > 1h 840 857 – – #basis : number of intermediate basis elements check : time for checking the Gröbner basis candidate in the trace algorithm monic : time for making the normal forms monic

• ld trace : trace algorithm (old implementation)

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 20

Example 2 : Cap

A multiple extension field Q(α1, α2) is defined by m1(t1) = t7

1 − 7t1 + 3,

m2(t1, t2) = t6

2 + t1t5 2 + t2 1t4 2 + t3 1t3 2 + t4 1t2 2 + t5 1t2 + t6 1 − 7,

m1(α1) = 0, m2(α1, α2) = 0. Cap = modified Caprasse, which contains α1, α2 non-trace trace #basis check monic

• ver Q(α1, α2)

294 306 45 242 20

• ver Q

— > 1h — – –

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 21

Example 3 : Cohn3|t=ω

ω is a root of md(t) = 3td + 2t + 2.

(CAUTION: This example causes an infinte loop in Asir in the

• DVD. It only accepts monic defining polynomials.)

d trace #basis check monic 2 over Q(ω) 1.6 32 1.4 2 over Q 0.5 85 – 5 over Q(ω) 12 32 9 5 over Q 21 253 – 10 over Q(ω) 70 32 44 10 over Q 304 447 –

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 22

Discussion

The new method is faster if the extension degree is high. The extension degree = 2 ⇒ the computation

• ver Q is faster than the new one (Example1,3).

Possible reasons : selection strategy, overhead by making monic Trace algorithm is efficient if the check is not hard. Example2 : the check is very hard because the result contains very large integer coefficients. Frequent inter-reduction is effective for reducing the total cost.

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 23

Possible improvements

Optimization of the interval of inter-reduction The default value (= 6) may not be optimized. F4 algorithm over algebraic number fields Current F4 implementation in Risa/Asir is not fully optimized for computation over algebraic number fields. ⇒ CRT for algebraic number fields should be implemented for solving system of linear equations.

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 24

Future work : Combination with dynamic evaluation It is often hard to make the defining ideal J maximal. A relaxation If J is a zero-dimensional radical ideal ⇒ h = HCR(r) may not be invertible ⇒ J = (J : h) ∩ (J + h) by dynamic evaluation ⇒ the Gröbner basis computation is also split. (Bases obtained before the splitting are valid because they are monic.)

An Efficient Implementation for Computing Gr¨

• bner bases over algebraic number fields – p. 25