Complexity ©D. Moshkovitz 1 And Randomized Computations The Polynomial Hierarchy.
On the complexity of computations modulo zero-dimensional …poteaux/fichiers/linz-trig-sets.pdf ·...
Transcript of On the complexity of computations modulo zero-dimensional …poteaux/fichiers/linz-trig-sets.pdf ·...
On the complexity of computations modulozero-dimensional triangular sets.
Adrien Poteaux? †, Éric Schost†
?: Calcul Formel - LIFL - Université Lille 1†: Computer Science Department, The University of Western Ontario, London, ON, Canada
GBReLA 2013, Hagenberg, Austria
Thursday, September 5th 2013
[email protected] Triangular sets 1 / 17
On the complexity of computations modulozero-dimensional triangular sets.
Adrien Poteaux, Éric Schost
Modular composition modulo triangular sets and applications
Computational Complexity, September 2013, Volume 22, Issue 3, pp 463-516
On the complexity of computing with zero-dimensional triangular sets
Journal of Symbolic Computation, Volume 50, March 2013, Pages 110-138
[email protected] Triangular sets 1 / 17
Triangular sets
K a field.
Y = Y1, . . . ,Ys variables on K, order Y1 < · · · < Ys .
Triangular set (monic, squarefree, of dimension 0):
T
∣∣∣∣∣∣∣Ts(Y1, . . . ,Ys)...
T1(Y1)
- Ti ∈ K[Y1, · · · ,Yi ] monic in Yi
- Ti reduced modulo 〈T1, . . . ,Ti−1〉.
Notations:- di = degYi
(Ti ) ≥ 2 ; d = (d1, . . . , ds) multidegree of T.- δ = d1 · · · ds
- RT = K[Y]/〈T〉 ' K[Y]d
[email protected] Triangular sets 1 / 17
One exampleC. Pascal & É. Schost 2006, Change of order for bivariate triangular sets
Aim: a factor of T1 = Y 6 − 5Y 5 + 6Y 4 − 9Y 3 + 6Y 2 − 5Y + 1.
Roots of T1 invariants by α 7→ 1α
T
∣∣∣∣∣ T2 = Y2 − (Y1 +1
Y1) mod T1 = Y2 − (Y 5
1 − 5Y 41 + 6Y 3
1 − 9Y 21 + 5Y1 − 5)
T1(Y1) = Y 61 − 5Y 5
1 + 6Y 41 − 9Y 3
1 + 6Y 21 − 5Y1 + 1
Change of order Y2 < Y1∣∣∣∣ Y 21 − Y2Y1 + 1
Y 32 − 5Y 2
2 + 3Y2 + 1
Y 32 − 5Y 2
2 + 3Y2 + 1 = (Y 22 − 4Y2 − 1)(Y2 − 1)
Back to the initial order∣∣∣∣ Y 21 − Y2Y1 + 1
Y 22 − 4Y2 − 1 =⇒
∣∣∣∣ Y2 + Y 31 − 4Y 2
1 − 4Y 4
1 − 4Y 31 + Y 2
1 − 4Y1 + 1
=⇒ degree of the polynomial /2
[email protected] Triangular sets 2 / 17
One exampleC. Pascal & É. Schost 2006, Change of order for bivariate triangular sets
Aim: a factor of T1 = Y 6 − 5Y 5 + 6Y 4 − 9Y 3 + 6Y 2 − 5Y + 1.Roots of T1 invariants by α 7→ 1
α
T
∣∣∣∣∣ T2 = Y2 − (Y1 +1
Y1) mod T1 = Y2 − (Y 5
1 − 5Y 41 + 6Y 3
1 − 9Y 21 + 5Y1 − 5)
T1(Y1) = Y 61 − 5Y 5
1 + 6Y 41 − 9Y 3
1 + 6Y 21 − 5Y1 + 1
Change of order Y2 < Y1∣∣∣∣ Y 21 − Y2Y1 + 1
Y 32 − 5Y 2
2 + 3Y2 + 1
Y 32 − 5Y 2
2 + 3Y2 + 1 = (Y 22 − 4Y2 − 1)(Y2 − 1)
Back to the initial order∣∣∣∣ Y 21 − Y2Y1 + 1
Y 22 − 4Y2 − 1 =⇒
∣∣∣∣ Y2 + Y 31 − 4Y 2
1 − 4Y 4
1 − 4Y 31 + Y 2
1 − 4Y1 + 1
=⇒ degree of the polynomial /2
[email protected] Triangular sets 2 / 17
One exampleC. Pascal & É. Schost 2006, Change of order for bivariate triangular sets
Aim: a factor of T1 = Y 6 − 5Y 5 + 6Y 4 − 9Y 3 + 6Y 2 − 5Y + 1.Roots of T1 invariants by α 7→ 1
α
T
∣∣∣∣∣ T2 = Y2 − (Y1 +1
Y1) mod T1 = Y2 − (Y 5
1 − 5Y 41 + 6Y 3
1 − 9Y 21 + 5Y1 − 5)
T1(Y1) = Y 61 − 5Y 5
1 + 6Y 41 − 9Y 3
1 + 6Y 21 − 5Y1 + 1
Change of order Y2 < Y1∣∣∣∣ Y 21 − Y2Y1 + 1
Y 32 − 5Y 2
2 + 3Y2 + 1
Y 32 − 5Y 2
2 + 3Y2 + 1 = (Y 22 − 4Y2 − 1)(Y2 − 1)
Back to the initial order∣∣∣∣ Y 21 − Y2Y1 + 1
Y 22 − 4Y2 − 1 =⇒
∣∣∣∣ Y2 + Y 31 − 4Y 2
1 − 4Y 4
1 − 4Y 31 + Y 2
1 − 4Y1 + 1
=⇒ degree of the polynomial /2
[email protected] Triangular sets 2 / 17
One exampleC. Pascal & É. Schost 2006, Change of order for bivariate triangular sets
Aim: a factor of T1 = Y 6 − 5Y 5 + 6Y 4 − 9Y 3 + 6Y 2 − 5Y + 1.Roots of T1 invariants by α 7→ 1
α
T
∣∣∣∣∣ T2 = Y2 − (Y1 +1
Y1) mod T1 = Y2 − (Y 5
1 − 5Y 41 + 6Y 3
1 − 9Y 21 + 5Y1 − 5)
T1(Y1) = Y 61 − 5Y 5
1 + 6Y 41 − 9Y 3
1 + 6Y 21 − 5Y1 + 1
Change of order Y2 < Y1∣∣∣∣ Y 21 − Y2Y1 + 1
Y 32 − 5Y 2
2 + 3Y2 + 1
Y 32 − 5Y 2
2 + 3Y2 + 1 = (Y 22 − 4Y2 − 1)(Y2 − 1)
Back to the initial order∣∣∣∣ Y 21 − Y2Y1 + 1
Y 22 − 4Y2 − 1 =⇒
∣∣∣∣ Y2 + Y 31 − 4Y 2
1 − 4Y 4
1 − 4Y 31 + Y 2
1 − 4Y1 + 1
=⇒ degree of the polynomial /2
[email protected] Triangular sets 2 / 17
One exampleC. Pascal & É. Schost 2006, Change of order for bivariate triangular sets
Aim: a factor of T1 = Y 6 − 5Y 5 + 6Y 4 − 9Y 3 + 6Y 2 − 5Y + 1.Roots of T1 invariants by α 7→ 1
α
T
∣∣∣∣∣ T2 = Y2 − (Y1 +1
Y1) mod T1 = Y2 − (Y 5
1 − 5Y 41 + 6Y 3
1 − 9Y 21 + 5Y1 − 5)
T1(Y1) = Y 61 − 5Y 5
1 + 6Y 41 − 9Y 3
1 + 6Y 21 − 5Y1 + 1
Change of order Y2 < Y1∣∣∣∣ Y 21 − Y2Y1 + 1
Y 32 − 5Y 2
2 + 3Y2 + 1
Y 32 − 5Y 2
2 + 3Y2 + 1 = (Y 22 − 4Y2 − 1)(Y2 − 1)
Back to the initial order∣∣∣∣ Y 21 − Y2Y1 + 1
Y 22 − 4Y2 − 1 =⇒
∣∣∣∣ Y2 + Y 31 − 4Y 2
1 − 4Y 4
1 − 4Y 31 + Y 2
1 − 4Y1 + 1
=⇒ degree of the polynomial /2
[email protected] Triangular sets 2 / 17
One example
C. Pascal & É. Schost 2006, Change of order for bivariate triangular sets
Aim: a factor of T1 = Y 6 − 5Y 5 + 6Y 4 − 9Y 3 + 6Y 2 − 5Y + 1.
Roots of T1 invariants by α 7→ 1α
Change of order Y2 < Y1
Y 32 − 5Y 2
2 + 3Y2 + 1 = (Y 22 − 4Y2 − 1)(Y2 − 1)
Back to the initial order
=⇒ degree of the polynomial /2
[email protected] Triangular sets 2 / 17
Problems we consider
Multiplication O (̃4sδ) Li, Moreno Maza & Schost 09
Quasi-inverse O (̃K sδ) Dahan, Moreno Maza, Schost & Xie 06
Change of order O (̃δ(ω+1)/2) Pascal & Schost 06 ; s = 2
Equiprojetable dec. (s log d)O(1)d sO(1)Szántó 97 ; non radical case, dim≥ 0
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We want: quasi-linear algorithms
[email protected] Objectives 3 / 17
Problems we consider
Multiplication O (̃4sδ) Li, Moreno Maza & Schost 09
Quasi-inverse O (̃K sδ) Dahan, Moreno Maza, Schost & Xie 06
Change of order O (̃δ(ω+1)/2) Pascal & Schost 06 ; s = 2
Equiprojetable dec. (s log d)O(1)d sO(1)Szántó 97 ; non radical case, dim≥ 0
I = 〈T1〉 ∪ · · · ∪ 〈Tn〉 ������
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We want: quasi-linear algorithms
[email protected] Objectives 3 / 17
Problems we consider
Multiplication O (̃4sδ) Li, Moreno Maza & Schost 09
Quasi-inverse O (̃K sδ) Dahan, Moreno Maza, Schost & Xie 06
Change of order O (̃δ(ω+1)/2) Pascal & Schost 06 ; s = 2
Equiprojetable dec. (s log d)O(1)d sO(1)Szántó 97 ; non radical case, dim≥ 0
I = 〈T1〉 ∪ · · · ∪ 〈Tn〉 ������
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We want: quasi-linear algorithms
[email protected] Objectives 3 / 17
Idea: univariate representation
Univariate representation U = (P,U, µ) of an ideal I :
ψU : K[X]/I → K[Z ]/〈P〉X1, . . . ,Xs 7→ U1, . . . ,Us
µ1X1 + · · ·+ µsXs ←[ Z.
Finding U ? → s bivariate steps (“mixed” representation)
=⇒ modular composition and power projection.
Total: O(s2C(δ))
[email protected] Strategy 4 / 17
Idea: univariate representation
Univariate representation U = (P,U, µ) of an ideal I :
ψU : K[X]/I → K[Z ]/〈P〉X1, . . . ,Xs 7→ U1, . . . ,Us
µ1X1 + · · ·+ µsXs ←[ Z.
Finding U ? → s bivariate steps (“mixed” representation)
=⇒ modular composition and power projection.
Total: O(s2C(δ))
[email protected] Strategy 4 / 17
Modular composition
Univariate case: F (G ) mod H.
Multivariate case: F (G1, · · · ,Gm) ∈ RT.
* f = (f1, . . . , fm) ∈ Nm ; f1 · · · fm = δ
* T, G1, · · · ,Gm ∈ RT, F ∈ K[X1, · · · ,Xm]f ,
Complexity denoted C(δ).
Matrix representation:...
...G0
1 ···G0m mod 〈T〉 · · · G f1−1
1 ···G fm−1m mod 〈T〉
......
δ×δ
∗
...F...
δ×1
[email protected] Main tools 5 / 17
Modular composition
Univariate case: F (G ) mod H.
Multivariate case: F (G1, · · · ,Gm) ∈ RT.
* f = (f1, . . . , fm) ∈ Nm ; f1 · · · fm = δ
* T, G1, · · · ,Gm ∈ RT, F ∈ K[X1, · · · ,Xm]f ,
Complexity denoted C(δ).
Matrix representation:...
...G0
1 ···G0m mod 〈T〉 · · · G f1−1
1 ···G fm−1m mod 〈T〉
......
δ×δ
∗
...F...
δ×1
[email protected] Main tools 5 / 17
Power projection
Univariate case: τ(G i mod H) with i < f .
Multivariate case: τ(G a11 · · ·G am
m mod 〈T〉); 0 ≤ ai < fi , i = 1, . . . ,m.
. f = (f1, . . . , fm) ∈ Nm
. T, G1, · · · ,Gm ∈ RT, τ : RT → K,
Transposed of the modular composition:
(· · · τ · · ·
)1×δ ∗
...
...G0
1 ···G0m mod 〈T〉 · · · G f1−1
1 ···G fm−1m mod 〈T〉
......
δ×δ
=⇒ Complexity C(δ)
[email protected] Main tools 6 / 17
Power projection
Univariate case: τ(G i mod H) with i < f .
Multivariate case: τ(G a11 · · ·G am
m mod 〈T〉); 0 ≤ ai < fi , i = 1, . . . ,m.
. f = (f1, . . . , fm) ∈ Nm
. T, G1, · · · ,Gm ∈ RT, τ : RT → K,
Transposed of the modular composition:
(· · · τ · · ·
)1×δ ∗
...
...G0
1 ···G0m mod 〈T〉 · · · G f1−1
1 ···G fm−1m mod 〈T〉
......
δ×δ
=⇒ Complexity C(δ)
[email protected] Main tools 6 / 17
C(δ) = ?
Algebraic model:
→ Brent & Kung 1978:
- m = s = 1, δ = d .
- C(d) = O(d(ω+1)/2)
→ Generalisation:
- m, s ∈ {1, 2}.- C(δ) = O(δ(ω+1)/2)
Idea:
1 “Divide” F as δ1/2 polynomials with degrees f 1/21 × f 1/2
2 ,
=⇒ compute G j11 G j2
2 , jk < f 1/2k . O(4sδ3/2).
2 Use matrix multiplication for the “small” blocks. O(δ(ω+1)/2)
3 Get the result via Horner. O(4sδ3/2)
[email protected] Modular composition 7 / 17
C(δ) = ?Boolean model: K = Fq ; binary complexity
→ Kedlaya & Umans 2011:- s = 1, f = (d , · · · , d), δ = N.
- C(dm,N) = (dm + N)1+ε log1+o(1)(q)
→ Generalisation:- m, s ∈ {1, 2}.- C(δ) = δ1+ε log(q) plog(log q)
Ideas:Modular composition ⇐⇒ Multivariate Multipoint Evaluation (MME)
1 Reformat the polynomial (more variables ; smaller degrees)
=⇒ composition then reduction.2 Compute the composition via evaluation - interpolation.
Fast structured evaluation and interpolation + 1 MME
Multivariate Multipoint Evaluation:1 Data considered in Z (or Z[Z ]) ; successive reductions modulo small p.2 p ' m f =⇒ evaluation at all points of Fm
p (FFT) then CRT.
[email protected] Modular composition 8 / 17
A theoretical algorithm (at least yet)
R. Basson & G. Lecerf: C++ implementation in Mathemagix
f (g(X )) mod h(X ) ; degX f , g , h < N
K = Fp, p ∼ 232
25 210 213N
µs
225
25
21
210
215
220
21
[email protected] Modular composition 9 / 17
Changing representation: s = 2
T = (T1,T2) in K[Y1,Y2] with degree d = (d1, d2)
car(K) ≥ δ2 or car(K) = 0
T squarefree ?gcd computation
Yes ? compute a primitive representation U
trace computations: power projection
Isomorphisms ?modular composition
=⇒ Cost: O(C(δ))
[email protected] Changing representation 10 / 17
Changing representation: general case
1 d ∈ Ns , T = (T1, . . . ,Ts) in K[Y], car(K) ≥ δ2 or car(K) = 0
〈T1,T2〉 radical ideal ? gcd computation
Yes ? compute a 2-mixed representation M of Ttrace computations: power projection + modular compositions
=⇒ O(s C(δ))
isomorphism computations: modular compositions
=⇒ O(C(δ))
2 “Repeat” s − 1 times: O(s2 C(δ))
[email protected] Changing representation 11 / 17
Operations modulo 〈T 〉
1 Case K = Fq: q ≥ δ ⇒ q′ = q ; q < δ ⇒ extension field
2 U = (P,U, µ) primitive representation of T: O(s2C(δ))
3 Operations in K[Y ]/〈P〉.
Multiplication: δ plog(δ)
Inversion: extended gcd δ plog(δ)
Norm: resultant δ plog(δ)
Modular composition: m ≤ 2 O(δ(ω+1)/2)
δ1+ε log(q′) plogε(log(q′))
Power projection: transposed algorithm O(δ(ω+1)/2)
δ1+ε log(q′) plogε(log(q′))
[email protected] Results 12 / 17
Equiprojetable decomposition / change of orderStrategy:
1 Compute a univariate representation U = (P,U, µ)
2 Bivariate case: O(C(δ) log(δ))
µ′ = µ′1Y1 + · · ·+ µ′s−1Ys−1 primitive elt (s − 1 first variables).
Inverse modular composition (traces ; power projections)
Compute its characteristic polynomial: χµ′ = C r11 · · ·C
rnn
Trace computations → power projection
Compute gcd(Ci (µ′1U1 + · · ·+ µ′s−1Us−1),P), 1 ≤ i ≤ n.
Recursive computation following the decompotition tree of χµ′
3 “Repeat” s times
Total cost: O(s C(δ) log(δ))
Remark: we assume car(K) ≥ δ[email protected] Results 13 / 17
Maple bench: us versus RegularChainsdi δ Us Maple2 3 .30e-1 .1193 6 .41e-1 .40e-14 10 .70e-1 .1195 15 .81e-1 .2696 21 .161 .699
di δ Us Maple2 4 .40e-1 .31e-13 10 .81e-1 .1404 20 .170 .5905 35 .330 1.7406 56 .520 4.980
s = 2 s = 3
di δ Us Maple2 5 .80e-1 .50e-13 15 .200 .3804 35 .510 2.2805 70 1.060 8.8006 126 2.510 39.450
di δ Us Maple2 6 .230 .1003 21 .370 1.0004 56 1.090 6.6605 126 3.230 45.2206 252 12.380 459.130
s = 4 s = 5
di δ Us Maple2 7 .360 .1603 28 .670 2.1704 84 2.490 19.6405 210 10.570 262.1006 462 62.940 6155.290
s = 6
[email protected] Results 14 / 17
Equiprojetable decomposition =⇒ modular composition
Hyp: m = 1 and s ≤ 2 ; T = (T1,T2) ∈ K[X1,X2] radical
E(n, δ) = complexity of the equiprojetable decomposition
G ∈ RT and F ∈ K[Y] given, K = F (G ) ∈ RT ?
=⇒ 2E(4, δ) + O~(δ)
s arbitrary can be generalised in 2E(s+ 2, δ) + O~(δ)
Equiprojetable decomposition =⇒ power projection
[email protected] Equivalent problems 15 / 17
Details
Let
T′∣∣∣∣∣∣
Y − G(X1,X2)T2(X1,X2)T1(X1)
with order X1 < X2 < Y
Change of order Y < X1 < X2:
U(i)
∣∣∣∣∣∣Ui,2(Y ,X1,X2)Ui,1(Y ,X1)Ri (Y )
�� ��Fi = F mod Ri ; order Y < X1 < X2 < Z .
Let I generated by:∣∣∣∣∣∣∣∣Z − F (Y )Y − G(X1,X2)T2(X1,X2)T1(X1)
I is actually generated by:
T′′
∣∣∣∣∣∣∣∣Z − K(X1,X2)Y − G(X1,X2)T2(X1,X2)T1(X1)
(order X1 < X2 < Y < Z)
But I is the intersection of the
V(i)
∣∣∣∣∣∣∣∣Z − Fi (Y )Ui,2(Y ,X1,X2)Ui,1(Y ,X1)Ri (Y )
E(3, δ)
[email protected] Equivalent problems 16 / 17
Details
Let
T′∣∣∣∣∣∣
Y − G(X1,X2)T2(X1,X2)T1(X1)
with order X1 < X2 < Y
Change of order Y < X1 < X2:
U(i)
∣∣∣∣∣∣Ui,2(Y ,X1,X2)Ui,1(Y ,X1)Ri (Y )
�� ��Fi = F mod Ri ; order Y < X1 < X2 < Z .
Let I generated by:∣∣∣∣∣∣∣∣Z − F (Y )Y − G(X1,X2)T2(X1,X2)T1(X1)
I is actually generated by:
T′′
∣∣∣∣∣∣∣∣Z − K(X1,X2)Y − G(X1,X2)T2(X1,X2)T1(X1)
(order X1 < X2 < Y < Z)
But I is the intersection of the
V(i)
∣∣∣∣∣∣∣∣Z − Fi (Y )Ui,2(Y ,X1,X2)Ui,1(Y ,X1)Ri (Y )
E(3, δ)
[email protected] Equivalent problems 16 / 17
Details
Let
T′∣∣∣∣∣∣
Y − G(X1,X2)T2(X1,X2)T1(X1)
with order X1 < X2 < Y
Change of order Y < X1 < X2:
U(i)
∣∣∣∣∣∣Ui,2(Y ,X1,X2)Ui,1(Y ,X1)Ri (Y )
�� ��Fi = F mod Ri ; order Y < X1 < X2 < Z .
Let I generated by:∣∣∣∣∣∣∣∣Z − F (Y )Y − G(X1,X2)T2(X1,X2)T1(X1)
I is actually generated by:
T′′
∣∣∣∣∣∣∣∣Z − K(X1,X2)Y − G(X1,X2)T2(X1,X2)T1(X1)
(order X1 < X2 < Y < Z)
But I is the intersection of the
V(i)
∣∣∣∣∣∣∣∣Z − Fi (Y )Ui,2(Y ,X1,X2)Ui,1(Y ,X1)Ri (Y )
E(3, δ)
[email protected] Equivalent problems 16 / 17
Details
Let
T′∣∣∣∣∣∣
Y − G(X1,X2)T2(X1,X2)T1(X1)
with order X1 < X2 < Y
Change of order Y < X1 < X2:
U(i)
∣∣∣∣∣∣Ui,2(Y ,X1,X2)Ui,1(Y ,X1)Ri (Y )
�� ��Fi = F mod Ri ; order Y < X1 < X2 < Z .
Let I generated by:∣∣∣∣∣∣∣∣Z − F (Y )Y − G(X1,X2)T2(X1,X2)T1(X1)
I is actually generated by:
T′′
∣∣∣∣∣∣∣∣Z − K(X1,X2)Y − G(X1,X2)T2(X1,X2)T1(X1)
(order X1 < X2 < Y < Z)
But I is the intersection of the
V(i)
∣∣∣∣∣∣∣∣Z − Fi (Y )Ui,2(Y ,X1,X2)Ui,1(Y ,X1)Ri (Y )
E(3, δ)
[email protected] Equivalent problems 16 / 17
Details: complexity
Let
T′∣∣∣∣∣∣
Y − G(X1,X2)T2(X1,X2)T1(X1)
with order X1 < X2 < Y
Change of order Y < X1 < X2:
U(i)
∣∣∣∣∣∣Ui,2(Y ,X1,X2)Ui,1(Y ,X1)Ri (Y )
�� ��Fi = F mod Ri ; order Y < X1 < X2 < Z .
Let I generated by:∣∣∣∣∣∣∣∣Z − F (Y )Y − G(X1,X2)T2(X1,X2)T1(X1)
I is actually generated by:
T′′
∣∣∣∣∣∣∣∣Z − K(X1,X2)Y − G(X1,X2)T2(X1,X2)T1(X1)
(order X1 < X2 < Y < Z)
But I is the intersection of the
V(i)
∣∣∣∣∣∣∣∣Z − Fi (Y )Ui,2(Y ,X1,X2)Ui,1(Y ,X1)Ri (Y )
E(3, δ)
[email protected] Equivalent problems 16 / 17
Details: complexity
Let
T′∣∣∣∣∣∣
Y − G(X1,X2)T2(X1,X2)T1(X1)
with order X1 < X2 < Y
Change of order Y < X1 < X2:
U(i)
∣∣∣∣∣∣Ui,2(Y ,X1,X2)Ui,1(Y ,X1)Ri (Y )
�� ��Fi = F mod Ri ; order Y < X1 < X2 < Z .
Let I generated by:∣∣∣∣∣∣∣∣Z − F (Y )Y − G(X1,X2)T2(X1,X2)T1(X1)
I is actually generated by:
T′′
∣∣∣∣∣∣∣∣Z − K(X1,X2)Y − G(X1,X2)T2(X1,X2)T1(X1)
(order X1 < X2 < Y < Z)
But I is the intersection of the
V(i)
∣∣∣∣∣∣∣∣Z − Fi (Y )Ui,2(Y ,X1,X2)Ui,1(Y ,X1)Ri (Y )
E(3, δ)
O(M(δ) log(δ))
[email protected] Equivalent problems 16 / 17
Details: complexity
Let
T′∣∣∣∣∣∣
Y − G(X1,X2)T2(X1,X2)T1(X1)
with order X1 < X2 < Y
Change of order Y < X1 < X2:
U(i)
∣∣∣∣∣∣Ui,2(Y ,X1,X2)Ui,1(Y ,X1)Ri (Y )
�� ��Fi = F mod Ri ; order Y < X1 < X2 < Z .
Let I generated by:∣∣∣∣∣∣∣∣Z − F (Y )Y − G(X1,X2)T2(X1,X2)T1(X1)
I is actually generated by:
T′′
∣∣∣∣∣∣∣∣Z − K(X1,X2)Y − G(X1,X2)T2(X1,X2)T1(X1)
(order X1 < X2 < Y < Z)
But I is the intersection of the
V(i)
∣∣∣∣∣∣∣∣Z − Fi (Y )Ui,2(Y ,X1,X2)Ui,1(Y ,X1)Ri (Y )
E(3, δ)
O(M(δ) log(δ))
E(4, δ)
[email protected] Equivalent problems 16 / 17
ConclusionComplexity quasi-linear / sub-quadratic for:
- multiplication, inversion, norm computation, modular composition andpower projection,
- change of order, equiprojetable decomposition.
Interesting practical results
Modular composition, power projectionm
Equiprojetable decomposition
Open questions:
- algebraic case: quasi-linear algorithm ?
- boolean case: algorithm usable in practice ?
- non radical ideals ?
- adaptation to the differentiel case ?
[email protected] Triangular sets 17 / 17