Structured FFT and TFT: symmetric and lattice polynomials∗

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Structured FFT and TFT:

symmetric and lattice polynomials

^∗

Joris van der Hoeven

Laboratoire d’informatique UMR 7161 CNRS École polytechnique 91128 Palaiseau Cedex, France

[email protected]

Romain Lebreton

LIRMM UMR 5506 CNRS Université de Montpellier II

Montpellier, France

[email protected]

Éric Schost

Computer Science Department Western University

London, Ontario Canada

[email protected]

ABSTRACT

In this paper, we consider the problem of efficient computations with structured polynomials. We provide complexity results for computing Fourier Transform and Truncated Fourier Transform of symmetric polynomials, and for mul- tiplying polynomials supported on a lattice.

1. INTRODUCTION

Fast computations with multivariate polynomials and power series have been of fundamental importance since the early ages of computer algebra. The representation is an important issue which conditions the performance in an intrinsic way; see [24,33,8] for some historical references.

It is customary to distinguish three main types of representations: dense, sparse, and functional. Adense represen- tation is made of a compact description of the support of the polynomial and the sequence of its coefficients. The main example concerns block supports – it suffices to store the coordinates of two opposite vertices. In a dense representation all the coefficients of the considered support are stored, even if they are zero. If a polynomial has only a few non-zero terms in its bounding block, we shall prefer to use asparse representation which stores only the sequence of the non- zero terms as pairs of monomials and coefficients. Finally, a functional representationstores a function that can produce values of the polynomials at any given point. This can be a pure blackbox (which means that its internal structure is not supposed to be known) or a specific data structure such asstraight-line programs (see Chapter 4 of [4], for instance, and [11] for a concrete library for the manipulation of polynomials with a functional representation).

For dense representations with block supports, it is classical that the algorithms used for the univariate case can be naturally extended: the naive algorithm, Karatsuba’s algorithm, and even Fast Fourier Transforms [7,32,6, 30, 17] can be applied recursively in each variable, with good performance. Another classical approach is the Kronecker

*This work has been partly supported by theDigiteo 2009-36HD grant of the Région Ile-de-France, the ANR grant HPAC (ANR-11- BS02-013), NSERC and the CRC program.

Permission to make digital or hard copies of all or part of this work for per- sonal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.

ISSAC’13, June 26–29, 2013, Boston, Massachusetts, USA.

substitution which reduces the multivariate product to one variable only; for all these questions, we refer the reader to classical books such as [29, 14]. When the number of variables is fixed and the partial degrees tend to infinity, these techniques lead to softly linear costs.

After the discovery of sparse interpolation [2, 25, 26, 27, 12], probabilistic algorithms with a quasi-linear complexity have been developed for sparse polynomial multiplication [5]. It has recently be shown that such asymptotically fast algorithms may indeed become more efficient than naive sparse multiplication [20].

In practice however, it frequently happens that multivariate polynomials with a dense flavor do not admit a block support. For instance, it is common to consider polynomials of a given total degree. In a recent series of works [16,18, 19, 22, 21], we have studied the complexity of polynomial multiplication in this “semi-dense” setting; see also [10,31].

In the case when the supports of the polynomials are ini- tial segments for the partial ordering onNⁿ, the truncated Fourier transform is a useful device for the design of efficient algorithms.

Besides polynomials with supports of a special kind, we may also consider what will call “structured polynomials”.

By analogy with linear algebra, such polynomials carry a special structure which might be exploited for the design of more efficient algorithms. In this paper, we turn our attention to a first important example of this kind: polynomials which are invariant under the action of certain matrix groups. We consider only two special cases: finite subgroups of S_n and finite groups of diagonal matrices. These cases are already sufficient to address questions raisede.g.in celes- tial mechanics [13]; it is hoped that more general groups can be dealt with using similar ideas.

In the limited scope of this paper, our main objective is to prove complexity results that demonstrate the savings induced by a proper use of the symmetries. Our complexity analyses take into account the number of arithmetic operations in the base field, and as often, we consider that operations with groups and lattices take a constant number of operations. A serious implementation of our algorithms would require an improved study of these aspects.

Of course, there already exists an abundant body of work on some of these questions. Crystallographic FFT algorithms date back to [34], with contributions as recent as [28], but are dedicated to crystallographic symmetries. A more general framework due to [1] was recently revisited under the point of view of high-performance code genera- tion [23]; our treatment of permutation groups is in a similar spirit, but to our understanding, these previous papers do

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not prove results such as those we give below (and they only consider the FFT, not its truncated version).

Also our results on diagonal groups, which fit in the general context of FFTs over lattices, use similar techniques as in a series of papers initiated by [15] and continued as recently as [35,3], but the actual results we prove are not in those references.

2. THE CLASSICAL FFT 2.1 Notation

We will work over an effective base field (or ring)Kwith sufficiently many roots of unity; the main objects are poly- nomialsK[x]⁸K[x1,, xn]innvariables overK. For any k= (k₁,,kn)∈Nⁿand P ∈K[x], we denote by x^kthe monomialx1k1

xnknand by Pkthe coefficient of P in x^k. The support supp(P)of a polynomialP∈K[x]is the set of exponentsk∈Nⁿsuch thatPk0.

For any subset S ofNⁿ, we define K[x]S as the polynomials with support included inS. As an important special case, whenS=N_dⁿ⁸{0,, d−1}ⁿ, we denote byK[x]d⁸

K[x]Sthe set of polynomials with partial degree less thand in all variables.

Letω∈Kbe a primitived-th root of unity (in Section4, it will be convenient to write this roote²^p^i/d). For anyk= (k1,,kn)∈Nⁿ, we define ω^k= (ω^k¹,, ω^kⁿ). One of our aims is to compute efficiently the map

FFTω: (

K[x]d K^N^dⁿ P (P(ωⁱ))i∈N_dⁿ

when P is a “structured polynomial”. Here, K^N^dⁿ denotes the set of vectors with entries inKindexed byN_dⁿ; in other words,v∈K^N^dⁿimplies thatvi∈Kfor alli∈N_dⁿ. Likewise, K^N^dⁿ^×N^dⁿdenotes the set of matrices with indices inNdn, that is,M∈K^N^dⁿ^×N^dⁿimplies thatMi,j∈Kfor alli,j∈N_dⁿ.

Most of the time, we will taked= 2^ℓwithℓ∈N, although we will also need more generaldof mixed radicesd=p1 pℓ. We denote byi·jthe inner product of two vectors inNⁿ. We also letek⁸(0,^k−1 ,0,1,0,,0), for16k6n, be thek- th element of the canonical basis ofKⁿ. At last, forℓ∈Nand k∈N₂ℓwe denote by[k]ℓthe bit reversal ofkin lengthℓand we extend this notation to vectors by[k]s= ([k1]s,,[kn]s).

2.2 The classical multivariate FFT

Let us first consider the in-place computation of a fulln- dimensional FFT of lengthd= 2^ℓin each variable. We first recall the notations and operations of the decimation in time variant of the FFT, and we refer the reader to [9] for more details. In what follows,ωis ad-th primitive root of unity.

We start with the FFT of a univariate polynomial P ∈ K[x]d. Decimation in time amounts to decomposingP into its even and odd parts, and proceeding recursively, by means ofℓdecimations applied to the variablex. For06k < d, we will writec_k⁰=Pkfor the input and denote byc^s= (cks)06k<d

the result aftersdecimation steps, fors∈ {1,, ℓ}.

At stages, the decimation is computed using butterflies of spanδ⁸2^ℓ−s. If i∈N₂sis even and jbelongs toN_δthen these butterflies are given by

c_iδ+j^s c_(i+1)δ+^s _j

!

= 1 ω^[i]^s^δ 1 −ω^[i]^s^δ

! c_iδ+j^s−1 c_(i+1)δ+j^s−1

! .

Putting the coefficients of all these linear relations in a matrix B^s∈K^N^d^×N^d, we getc^s=B^sc^s−1. The matrixB^s is sparse, with at most two non-zero coefficients on each row and each column; up to permutation, it is block-diagonal, with blocks of size 2. After ℓ stages, we get the evaluations of P in the bit reversed order:c_k^ℓ=P(ω^[k]^ℓ).

We now adapt this notation to the multivariate case. The computation is still divided inℓstages, each stage doing one decimation in every variable x1,, xn. Therefore, we will denote by ck0

8 Pkthe coefficients of the input fork∈N_d and by c_k^s,tthe coefficients obtained during the s-th stage, after the decimations inx1,, xtare done, withs∈ {1,, ℓ}

and t∈ {0,, n}, so that c_k^s−1,n= c_k^s,0. We abbreviate c_k^s−1⁸c_k^s,0for the coefficients afters−1stages.

The intermediate coefficientsc_k^scan be seen as evaluations of intermediate polynomials: for everys∈ {0,, ℓ},i∈N₂ns

and j∈Nδn, one has

c_iδ+j^s = Pjs((ω^δ)^[i]^s) (1) where Pjs=P

i^′∈N2sn Pi^′δ+jxⁱ^′are obtained through an s- fold decimation ofP. Equivalently, the coefficientsckssatisfy

c_iδ+j^s = X

i^′∈N2sn

Pi^′δ+jω^[i]^s^·i^′^δ. (2) Thus,c^ℓ_j=P(ω^[j]^ℓ)yields FFTω(P)in bit reversed order.

For the concrete computation of the coefficients c_k^s at stages from the coefficients c_k^s−1 at stage s−1, we usen so called “elementary transforms” with respect to each of the variables x1,, xn. For any t ∈ {1,, n}, the coef- ficientsc_k^s,tare obtained from the coefficientsc_k^s,t−1through butterflies of span δ = 2^ℓ−s with respect to the variable xt; this can be rewritten by means of the formula

c_iδ+j^s,t c_(i+e

t)δ+j s,t

!

= 1 ω^[i^t^]^s^δ 1 −ω^[i^t^]^s^δ

! c_iδ+^s,t−1_j c_(i+e

t)δ+j s,t−1

! (3) for any i∈N₂ns with even t-th coordinate it and j ∈Nδn. Equation (3) can be rewritten more compactly in matrix form c^s,t = B^s,t c^s,t−1 where B^s,t is a sparse matrix in K^N^dⁿ^×N^dⁿwhich is naturally indexed by pairs of multi-indices in N_dⁿ. Setting B^s=B^s,nB^s,1∈K^N^dⁿ^×N^dⁿ,we also obtain a short formula forc^sas a function of c^s−1:

c^s = B^sc^s⁻¹ (4)

Remark that the matricesB^s,1,,B^s,n commute pairwise.

Notice also that each of the rows and columns ofB^s,thas at most 2 non zero entries and consequently those of B^s contains at most 2ⁿ non zero entries. For this reason, we can apply the matrices B^s,t (resp. B^s) within O(|Ndn|) = O(dⁿ)(resp. O(n dⁿ)) operations in K. Finally, the fulln- dimensional FFT of lengthd= 2^ℓcostsF(d, n)⁸3/2ℓ n dⁿ= 3/2dⁿlog(dⁿ)operations inK(see [14, Section 8.2]).

Example 1. Let us make these matrices explicit for polynomials in n= 2 variables and degree less than d= 4, so that ℓ= 2. We start with the first decimation inx1whose butterflies of spanδare captured byB^s,twithδ= 2^ℓ−s= 2¹, s= 1andt= 1. It takes as inputc_k⁰=c_k^1,0⁸Pkand outputs c_k^1,1for k∈N₄²={0,,3}². For any j1∈N₂={0,1}and k2∈ {0,,3}, we let

c_(j^1,1₁_,k₂₎ c_(2+j^1,1 ₁_,k₂₎

!

=

1 1 1 −1

c_(j^1,0₁_,k₂₎ c_(2+j^1,0 ₁_,k₂₎

! .

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So, B^1,1 is a 16 by 16 matrix, which is made of diagonal blocks of¹₁ ₋₁¹in a suitable basis. The decimation inx2

during stage1is similar : c_(k^1,2₁_,j₂₎ c_(k^1,2₁_,2+j₂₎

!

=

1 1 1 −1

c_(k^1,1₁_,j₂₎ c_(k^1,1₁_,2+j₂₎

!

for j2∈ N₂ and k1 ∈ N₄. Consequently, B^1,2 is made of diagonal blocks¹₁ ₋₁¹ (in another basis thanB^1,1). Their productB¹⁸B^1,2B^1,1corresponds to the operations







c_(j11,j2)

c_(2+j1 1,j2)

c_(j11,2+j2)

c_(2+j1 1,2+j2)







=







1 1 1 1

1 −1 1 −1 1 1 −1 −1 1 −1 −1 1













c_(j01,j2)

c_(2+j0 1,j2)

c_(j01,2+j2)

c_(2+j0 1,2+j2)







forj₁,j₂∈N₂. ThusB¹is made of such 4 by 4 matrices on the diagonal (in yet another basis). Note that in general, the matricesB^s,tandB^sare still made of diagonal blocks, but these blocks vary along the diagonal.

We can sum it all up in the two following algorithms.

Algorithm Butterfly

Input:n, degreed, stages, index(2i^′,j)of the butterfly with i^′∈N₂ns−1, j ∈N_δⁿ, root of unity ω and coefficients c∈K^N²ⁿfor the butterfly (cb⁸c_(2is−1′+b)δ+j forb∈N₂ⁿ).

Output: the output of the butterflyd∈K^N²ⁿ

Fort= 1,, ndo //decimation inxt

Forb∈N₂ⁿ={0,1}ⁿsuch thatbt= 0do

r=ω^[2i^t^′^]^s^δc_b+e_t //butterfly in one variable db=cb+r

db+et=cb−r Return db

Algorithm FFT

Input: n, degreed,ωand coefficientsc⁰∈K^N^dⁿof P Output: coefficientsc^ℓ∈K^N^dⁿin bit reversed order

Fors= 1,, ℓdo //stages

δ⁸d/2^s

Fori^′∈N₂ns−1,j∈N₂nℓ−sdo //pick a butterfly Forb∈N₂ⁿdo c_b⁸c_(2is−1′+b)δ+j

db=Butterfly(n, d, s,i^′,j,cb) Forb∈N₂ⁿdo c_(2is ′+b)δ+j

8d_b Return c^ℓ

In the last section, we will also use the ring isomorphism K[x]d/(x1d−1,, xnd−1) → [K[x]δ/(x1δ−1,, xnδ−1)]²^ns

P Q^s= (Qis(x))i∈N2sn

with Qis(x)⁸ P

j∈Nδn

P

i^′∈N2sn Pi^′δ+jω^i·i^′^δ

(ωⁱx)^j for i ∈ N₂ns, and (ωⁱ x)^j = (ωⁱ¹ x1)^j¹ (ωⁱⁿ xn)^jⁿ. These polynomials generalize the decomposition of a univariate polynomialsP intoPmod(x^d/2−1)andPmod(x^d/2+ 1).

We could obtain them through a decimation in frequency, but it is enough to remark that we can reconstruct them from the coefficientsc^sthanks to the formula

Qis(x) = X

j∈Nδn

c_[i]^s _s_δ+j(ωⁱx)^j. (5)

3. THE SYMMETRIC FFT

In this section, we letG⊆S_n be a permutation group.

The groupGacts on the polynomialsK[x]via the map ϕg:

( K[x] K[x]

P(x1,, xn) P^g(x)⁸P(xg(1),, xg(n)). We denote byK[x]^G⁸stabGK[x] the set of polynomials invariant under the action of G. Our main result here is that one can save a factor (roughly) |G|when computing the FFT of an invariant polynomial. Our approach is in the same spirit as the one in [1], where similar statements on the savings induced by (very general) symmetries can be found.

However, we are not aware of published results similar to Theorem7below.

3.1 The bivariate case

Let P ∈K[x₁, x2] be a symmetric bivariate polynomial of partial degrees less than d= 8, so that ℓ= 3; let also ω∈Kbe a primitive eighth root of unity. We detail on this easy example the methods used to exploit the symmetries to decrease the cost of the FFT.

The coefficients of P are placed on a 8 × 8 grid. The bivariate classical FFT consists in the application of butterflies of sizeδ⁸2^ℓ−sforsfrom1to3, as in Figure1. When some butterflies overlap, we draw only the shades of all but one of them. The result of stages= 3is the set of evaluations (P(ωⁱ¹, ωⁱ²))i∈N8

2in bit reversed order.

Figure 1. Classical bivariate FFT

We will remark below that each stagespreserves the sym- metry of the input. In particular, since our polynomial P is symmetric, the coefficients c_k¹ at stage 1are symmetric too, so we only need to compute the coefficientsc_(k1 1,k2)for (k1,k2)in thefundamental domain(sometimes called asym- metric unit)06k26k1<8. We choose to compute at stage s only the butterflies which involves at least an element of F; the set of indices that are included in a butterfly of span δ= 2^ℓ−sthat meetsF will be called the extensionFδof F.

Figure 2. Symmetric bivariate FFT

Every butterfly of stage1meets the fundamental domain, so we do not save anything there. However, we save 4out of 16 butterflies at stage2and6out of 16 butterflies at the third stage. Asymptotically in d, we gain a factor two in the number of arithmetic operations inKcompared to the classical FFT; this corresponds to the order of the group.

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3.2 Notation

The groupGalso acts onNⁿwithg(i) = (ig(1),,ig(n)) for any i∈Nⁿ. This action is consistent with the action on polynomials sinceϕg(x^k) =x^g⁻¹^(k). BecauseGacts on polynomials, it acts on the vector of its coefficients. More generally, ifI⊆Nⁿandv∈K^G(I), we denote byv^g∈K^Ithe vector defined byv_i^g=vg(i)for alli∈I. IfI⊆Nⁿis stable by G,i.e. G(I)⊆I, we define the set of invariant vectors K^{I ,G}byK^{I ,G}⁸{v∈K^I:∀g∈G,v^g=v}.

For any two setsI , J satisfyingJ⊆I⊆Nⁿ, we define the restriction vJ∈K^J of v∈K^I by (vJ)j=vj for all j∈J.

Recall the definition of the lexicographical order6_lex : for alli,j∈Nⁿ,i<lexj if there existsk∈ {0,, n−1}such thatit=jt for any16t6k and i_k+1<j_k+1. Finally for any subset S ⊆Nⁿ stable by G, we define a fundamental domainS^Gof the action of GonS byS^G⁸{i∈S:∀g∈G, i>_lexg(i)}together with the projectionπ^G:Nⁿ→Nⁿsuch as{π^G(i)}⁸G({i})∩(Nⁿ)^G.

3.3 Fundamental domains

LetF=F^[d]⁸(Ndn)^Gbe the fundamental domain associ- ated to the action ofGonN_dⁿ. AnyG-symmetric vectorc∈ K^N^dⁿcan be reconstructed from its restrictioncF∈K^F using the formulaci= (cF)π^G(i). As it turns out, the in-place FFT algorithm from Section2.2admits the important property that the coefficients at all stage are stillG-symmetric.

Lemma2. LetP∈K[x]dand the vectorsc⁰,,c^ℓ∈K^N^dⁿ be as in Section 2.2. Then ifP∈K[x]dG,c⁰,,c^ℓ∈K^N^dⁿ^,G. Proof. GivenP∈K[x]dG,k∈Ndnand g∈G, we clearly havec_g(k)⁰ =c_k⁰. For anys∈ {0,, ℓ},i,i^′∈N₂ns, j∈N₂nℓ−s

and g ∈ G ⊆ S_n, we also notice that g(i 2^ℓ−s + j) = g(i) 2^ℓ−s+ g(j),[g(i)]s=g([i]s) and g(i)·g(i^′) =i·i^′. Hence, using Equation (2), we get

cs_g(i2ℓ−s+j) = cs_g(i)2ℓ−s+g(j)

= X

i^′∈N2sn

Pi^′2^ℓ−s+g(j)ω^[g(i)]^s^·i^′²^ℓ−s

= X

i^′∈N2sn

Pg(i^′)2^ℓ−s+g(j)ω^[g(i)]^s^·g(i^′⁾²^ℓ−s

= X

i^′∈N2sn

Pg(i^′2^ℓ−s+j)ω^g([i]^s^)·g(i^′⁾²^ℓ−s

= X

i^′∈N2sn

Pi^′2^ℓ−s+jω^[i]^s^·i^′²^ℓ−s

= c_i2s ℓ−s+j.

Thus,c^s_g(k)=cks for allk∈N_dⁿ, whencec⁰,,c^ℓ∈K^N^dⁿ^,G. This lemma implies that for the computation of the FFT of aG-symmetric polynomialP, it suffices to compute the projectionsc_F⁰,,c_F^ℓ.

In order to apply formula (4) for the computation ofcFs as a function of c_F^s−1, it is not necessary to completely recon- structc^s−1, due to the sparsity of the matrixB^s. Instead, we define theδ-expansionof the setFbyF={k⊞ǫδ:k∈F , ǫ∈N₂ⁿ}, wherei⊞jstands for the “bitwise exclusive or” ofi andj. For anyk∈N_dⁿ, the set{k⊞ǫδ:ǫ∈N₂ⁿ}describes the vertices of the butterfly of spanδthat includes the pointk.

Thus,Fδis indeed the set of indices ofc^s−1that are involved in the computation of c^svia the formulac^s=B^sc^s−1.

Lemma 3. For any δ ∈ {1, 2, 4, , 2^ℓ−1}, let F[δ] = {k⊞ǫ:k∈F ,ǫ∈N_δⁿ},so that Fδ⊆F[2δ]. Then we have F[δ]=δ F^[d/δ]+N_δⁿ.

Proof. Assume that i∈F^[d/δ]. Then clearly, δi∈F, whence δi+N_δⁿ⊆F[δ]. Conversely, if i∈F[δ], then there exists a j∈F with j=i⊞ǫand ǫ∈Nδn. Letk=⌊j/δ⌋.

For any g∈G, we have g(k) =⌊g(j)/δ⌋6_lex⌊j/δ⌋=k.

Consequently,⌊j/δ⌋ ∈F^[d/δ], whence i∈δ⌊j/δ⌋+Nδn. For the proof of the next lemma, for(j , k)∈ {1,, n}², define∆j ,k={i∈Nⁿ:ij=ik}andΥ =N_dⁿ\S

kj∆j ,k. Lemma 4. There exists a constant C (depending on n) such that

|F^[d]| −_|G|^dⁿ

6C dⁿ⁻¹.

Proof. With the notation above, we have|∆j ,k∩N_dⁿ|= dⁿ⁻¹. TakingC=ⁿ₂, it follows that

S

kj∆j ,k

∩N_dⁿ 6 C dⁿ⁻¹. On the other hand, the orbit of any element inΥ underGcontains exactly|G|elements and one element inF. In other words,|F∩Υ|=|Υ|/|G|and so06|F| − |Υ|/|G|6 C dⁿ⁻¹. Finally, 06(dⁿ− |Υ|)/|G|6C/|G|dⁿ⁻¹ which implies−C/|G|dⁿ⁻¹6|F| −dⁿ/|G|6C dⁿ⁻¹.

3.4 The symmetric FFT

Letc_F^s−1_δ be the restriction of c^s−1 toFδ and notice that K^F^δ is stable underB^s,i.e. B^s(K^F^δ)⊆K^F^δ. Therefore the restrictionB_K^s^Fδof the mapB^s:K^N^dⁿ→K^N^dⁿto vectors inK^F^δ is aK-algebra morphism fromK^F^δtoK^F^δ. By construction, we now havecFs_δ=B_K^s^Fδc_F^s−1_δ .This allows us to computecFs

as a function of c_F^s−1using

c_F^s = (B_KsFδξδ(cFs−1))F (6) where ξδ(c_F^s−1)denotes theδ-expansionc_F^s−1_δ of c_F^s−1.

Remark 5.We could have saved a few more operations by only computing the coefficientsc_F^s. To do so, we would have performed just the operations of a butterfly corresponding to vertices inside F. However, the potential gain of complexity is only linear in the numbers of monomialsdⁿ.

The formula (6) yields a straightforward way to compute the direct FFT of aG-symmetric polynomial:

Algorithm Symmetric-FFT

Input:n,dand coefficientscF0 ∈K^N^dⁿof P∈K[x]dGinF Output:c_F^ℓ∈K^N^dⁿin bit reversed order

Fors= 1,, ℓdo δ⁸d/2^s

Fori^′∈N₂ns−1,j∈N₂nℓ−ss.t. 2i^′δ+j∈Fδdo Forb∈N₂ⁿdocb⁸c_πG((2i^′+b)δ+j)

s−1

db=Butterfly(n, d, s,i^′,j,cb) Forb∈N₂ⁿdocπs^G((2i^′+b)δ+j)

8db

Returnc^ℓ

Remark 6.The main challenge for an actual implementation of our algorithms consists in finding a way to iterate on sets likeFδwithout too much overhead. We give a hint on how to iterate overF in Lemma8(see also [23] for similar considerations).