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


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


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


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



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



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

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

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.


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


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)


Details: complexity

Let

T′∣∣∣∣∣∣

Y − G(X1,X2)T2(X1,X2)T1(X1)



U(i)

∣∣∣∣∣∣Ui,2(Y ,X1,X2)Ui,1(Y ,X1)Ri (Y )




T′′

∣∣∣∣∣∣∣∣Z − K(X1,X2)Y − G(X1,X2)T2(X1,X2)T1(X1)

(order X1 < X2 < Y < Z)


V(i)


E(3, δ)


Details: complexity

Let

T′∣∣∣∣∣∣

Y − G(X1,X2)T2(X1,X2)T1(X1)



U(i)

∣∣∣∣∣∣Ui,2(Y ,X1,X2)Ui,1(Y ,X1)Ri (Y )




T′′

∣∣∣∣∣∣∣∣Z − K(X1,X2)Y − G(X1,X2)T2(X1,X2)T1(X1)

(order X1 < X2 < Y < Z)


V(i)


E(3, δ)

O(M(δ) log(δ))


Details: complexity

Let

T′∣∣∣∣∣∣

Y − G(X1,X2)T2(X1,X2)T1(X1)



U(i)

∣∣∣∣∣∣Ui,2(Y ,X1,X2)Ui,1(Y ,X1)Ri (Y )




T′′

∣∣∣∣∣∣∣∣Z − K(X1,X2)Y − G(X1,X2)T2(X1,X2)T1(X1)

(order X1 < X2 < Y < Z)


V(i)


E(3, δ)

O(M(δ) log(δ))

E(4, δ)


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 ?


On the complexity of computations modulo zero-dimensional …poteaux/fichiers/linz-trig-sets.pdf ·...

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