Stable normal forms for polynomial system solving

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Theoretical Computer Science

journal homepage:www.elsevier.com/locate/tcs

Stable normal forms for polynomial system solving

Bernard Mourrain

a,∗

, Philippe Trébuchet

b

a_{GALAAD, INRIA Méditerranée, 2004 Route des Lucioles, BP 93, 06902 Sophia Antipolis, Cedex, France} b_{UPMC, LIP6, Equipe APR, 4 place jussieu, 75015 Paris, France}

a r t i c l e i n f o Keywords: Multivariate polynomial Quotient algebra Normal form Border basis Root-finding Symbolic-numeric computation a b s t r a c t

The paper describes and analyzes a method for computing border bases of a zero-dimensional idealI. The criterion used in the computation involves specific commutation polynomials, and leads to an algorithm and an implementation extending the ones in [B. Mourrain, Ph. Trébuchet, Generalised normal forms and polynomial system solving, in: M. Kauers (Ed.), Proc. Intern. Symp. on Symbolic and Algebraic Computation, ACM Press, New-York, 2005, pp. 253–260]. This general border basis algorithm weakens the monomial ordering requirement for Gröbner bases computations. It is currently the most general setting for representing quotient algebras, embedding into a single formalism Gröbner bases, Macaulay bases and a new representation that does not fit into the previous categories. With this formalism, we show how the syzygies of the border basis are generated by commutation relations. We also show that our construction of normal form is stable under small perturbations of the ideal, if the number of solutions remains constant. This feature has a huge impact on practical efficiency, as illustrated by the experiments on classical benchmark polynomial systems, at the end of the paper.

©2008 Elsevier B.V. All rights reserved. 1. Introduction

Solving polynomial systems is the cornerstone of many computations in robotics, geometric modeling, signal processing, chromatology, structural molecular biology etc. In these applications, the systems have, most of the time, finitely many solutions and the equations often appear with approximate coefficients.

From a computational point of view, it is a challenge to develop efficient and numerically stable methods for solving such systems. Backward stability is required, that is, the computed solution should be the exact solution of a system in a small neighborhood of the input. Efficiency is also mandatory to tackle the systems. One can expect, for instance, that the behavior of the method depends mainly on the number of solutions, and partially on other extrinsic parameters, such as the number of variables.

To ensure backward stability, one can apply classical numerical methods such as Newton-like iterations. These local methods, however, provide neither guarantee of global convergence nor a complete description of all the roots. Algebraic methods, on the contrary, handle all the roots simultaneously. They reduce the problem to computing the structure of the quotient algebraAof the polynomial ring modulo, the idealIgenerated by the input system [8]. Such a structure is given by a basisBof the algebraAas a vector space and by the tables of multiplication in the algebraA. Equivalently, it can be described by an algorithm that projects the ring of polynomials_K

[

x

]

onto the vector space

h

B

i

, generated by the basisB along the idealI. We call such a projection, a normal form for the idealI.

From a knowledge of multiplication tables, we deduce either an exact encoding of the roots through a rational univariate representation [32,12], or a numerical approximation by means of eigenvector computation [1,24,34]. Since the

∗_{Corresponding author.}

E-mail addresses:[email protected](B. Mourrain),[email protected](P. Trébuchet). 0304-3975/$ – see front matter©2008 Elsevier B.V. All rights reserved.

eigencomputation is a numerically well-controlled process [13], the remaining challenge is to compute efficiently and in a stable way the quotient algebra structureA.

In the family of algebraic methods, resultant-based techniques (see, e.g. [19,9,3]) exploit the properties of the coefficients matrices of the monomial multiples in the input equations. The multiplication tables can be expressed via Schur complements in these matrices [9]. Their construction is deeply linked to the geometry of the underlying variety, which makes these methods very tolerant to a small perturbation. Unfortunately they heavily rely on genericity hypotheses, and this reduces their applicability. Moreover, the size of the constructed matrices usually grows exponentially with the number of variables.

To avoid such a pitfall, the so-called H-bases have been studied [20,22]. The construction proceeds degree by degree, and stops when the terms of the highest degree of the computed polynomials generate the terms of the highest degree in all the idealI. The stopping criterion requires the computation of generators of the syzygies of these highest degree terms. Though it also yields a basis of the quotient ringA, without any apriori knowledge of its dimension, practically speaking, it also suffers from a swelling of the size of the linear systems to be solved.

The approach can be refined further, by using a grading in which the highest term of a polynomial is a monomial. This leads to Gröbner basis computation (see, e.g. [6]). The approach also yields a basis of the quotient algebraAand a normal form for the idealI. As in the other methods, their computation can be seen just as a triangulation of a certain matrix. Unfortunately these methods suffer from unavoidable instability: the monomial ordering attached to the Gröbner basis makes the pivot selection strategy in the triangulation depend only on the symbolic structure of the rows of the matrix, and not on the numerical values of the coefficients appearing in these rows. Thiscanlead to unwanted artificial singularities in the representation of the quotient ring. (Compare, for instance, the degree reverse lexicographic Gröbner basisp1

=

ax21

+

bx

2

2

+

ε

1x1x2,p2

=

cx21

+

dx 2

2

+

ε

2x1x2with

ε

1

=

ε

2

=

0 and

ε

1

6=

0,

ε

2

6=

0.)

To circumvent this artificial difficulty, a new approach was proposed in [25], based on a normal form criterion and involving commutation relations (for basesBconnected to 1). Further investigations of this technique in [27,28] and the Ph.D. dissertation [36] have led to a first version [29] of a normal form algorithm, which allowed one to construct efficiently and in a stable way the zero-dimensional quotient algebra representations. Similar investigations were also pursued in [15,

16], but in the more restrictive case, where the basis is stable by division. Other investigations related to the stabilization of the normal form process can be found in [34].

In this paper, we discuss border basis algorithms in the general sense, that is, for the basesBconnected to 1. See [26] for an introductory presentation of their properties. As in [11] or [10], our approach is based on the linear algebra tools. As for the H-bases, we use a grading of the polynomial ring, but the construction is optimized in the spirit of [27]. We describe an efficient criterion based on commutation polynomials to check the normal form property. This leads to an algorithm, presented in [29], which has been improved to treat optimally the case when a syzygy of the components of highest degree is found, which is not a syzygy of the corresponding polynomials. We prove that the syzygies of a (general) border basis are generated by the commutation relations, giving a short and concise answer to a conjecture in [16] for bases stable by division. Meanwhile, work related to the conjecture for this special case was also investigated in [14]. Regarding the numerical stability of border bases, we prove that our construction of normal form is stable against small perturbations of the input system, if the number of solutions remains constant.

The paper is structured as follows. The next section is devoted to the notations, and Section3to the stopping criterion for generalized normal forms. In Section4, we show how to recover the syzygies from the commutation relations. In Section5, we recall briefly the main idea of the algorithm in [29]. In Section6, we analyze the stability of this algorithm from a symbolic-numeric perspective. Finally, we show the efficiency of our implementation and its numerical behavior on classical polynomial benchmarks.

2. Notations

We recall some definitions in [25,27,28] and add a few more ones.

Let_Kbe an effective field. The ring ofn-variate polynomials over_Kwill be denoted byR,R

=

_K

[

x

] =

_K

[

x1

, . . . ,

xn

]

. We

considern-variate polynomialsf1

, . . . ,

fs

∈

R. These polynomials generate an ideal ofK

[

x

]

that we denoteI. The quotient of K

[

x

]

moduloIwill be denoted byA. From now on, we suppose thatI is zero dimensional, so thatAis a finite dimensional K-vector space. Our goal is to solve the system of equationsf1

=

0

, . . . ,

fs

=

0 over the algebraic closureKofK. The roots,

with coordinates in the algebraic closure of_K, are denoted

ζ

1

, . . . , ζ

d, with

ζ

i

=

(ζ

i,1

, . . . , ζ

i,n

)

∈

Kn.

The support supp

(

p

)

of a polynomialp

∈

_K

[

x

]

is the set of monomials appearing inpwith non-zero coefficients. Given a setSof elements of_K

[

x

]

, we denote by

h

S

i

the_K-vector space spanned by the elements ofS. Finally, we denote the set of all the monomials in the variablesx

=

(

x1

, . . . ,

xn

)

byM. A term is an element of the form

λ

·

mwith

λ

∈

K

− {

0

}

and

m

∈

M. For a subsetSofM, we denote bySc_{the set-theoretical complement of the setSinM. For any monomialm}

_∈

_M,

m

·

Mdenotes the set of all monomial multiples ofm.

For any subsetSofR, we denote byS+_orD

₍

_S

₎

_{the setS}+

₌

_S

_∪

_x

1S

∪ · · · ∪

xnS,

∂

S

=

S+

\

S.S+is called theprolongation

ofS. For anyk

∈

_N,S[k]_isSk+···+times_{the result of applyingktimes the operator}+_{onS. By convention,S}[0]_isS.

IfB

⊂

Mcontains 1, then for any monomialm

∈

Mthere existsksuch thatm

∈

B[k]_{. We say that a monomialmis of}

A set of monomialsBis said to be connected to 1 if and only if, 1

∈

Band for every monomialminB, there exists a finite sequence of variables

(

xij

)

j∈[1,l]such that

∀

l

0

_{∈ [}

₁

_,

_l

_]

_,

Q

j∈[1,l0_]xij

∈

Band

Q

j∈[1,l] xij

=

m.

A set of monomialsBis stable by division if for anym

∈

Band any variablexisuch thatm

=

xim0(m0

∈

M), we havem0

inB. Remark that a setBstable by division is connected to 1.

Definition 2.1. LetΛbe a monoid with a well order relation

≺

, such that:

∀

α, β, γ

∈

Λ

, α

≺

β

⇒

γ

+

α

≺

γ

+

β.

A

(

Λ

,

≺

)

-grading of_K

[

x

]

is the decomposition of_K

[

x

]

as the direct sum_K

[

x

] =

L

λ∈ΛK

[

x

]

[λ], with the following property:

∀

f

∈

_K

[

x

]

[α]

,

g

∈

K

[

x

]

[β]

⇒

f g

∈

K

[

x

]

[α+β]

.

The following element ofΛis denoted deg_Λ

(

f

)

or deg

(

f

)

(when no confusion is possible) and byΛ

(

f

)

, and is said to be the degree off: Λ

(

f

)

=

min

(

λ

∈

Λ

|

f

∈

M

λ0_λ K

[

x

]

[_λ0_]

)

.

For any setV

⊂

_K

[

x

]

, letV_λ

=

L

λ0_{λ K}

[

x

]

[_λ0_]

∩

V. For any

λ

∈

Λ, let

λ

+

=

min

{

λ

0

∈

Λ

;

_K

[

x

]

+_λ

⊂

_K

[

x

]

_λ0

}

and let

λ

−

₌

_max

_{

_λ

0

_∈

_Λ

_;

K

[

x

]

+_λ0

⊂

K

[

x

]

λ

}

.

In order not to be confused with different notions of degree, for any monomialmofK

[

x

]

, we define the size ofm, denoted

by

|

m

|

, to be the integerdsuch thatm

=

xi1

· · ·

xid, which itself imposes a grading.

Another classical grading, is the one associated with a monomial ordering, whereΛ

=

_Nn_and

_≺

_{is a monomial order}

(see [7, p. 328]) such that for all

α

=

(α

₁

, . . . , α

_n

)

∈

_Nn_{we have}

K

[

x

]

[α]

=

Kxα11

· · ·

xαnn

.

Definition 2.2. We say thatΛis a reducing grading, ifΛis a grading and if the following property holds: for all monomials m

,

m0

_∈

_{Msuch thatm}0_{strictly dividesmwe haveΛ}

₍

_m0

₎

_≺

_Λ

₍

_m

_),

_Λ

₍

_m0

₎

₆₌

_Λ

₍

_m

_).

Both gradings induced by the classical degree and a monomial ordering are reducing grading. Hereafter,we denote byΛa reducing grading.

Definition 2.3. A rewriting familyFfor a monomial setBis a set of polynomialsF

= {

fi

}

i∈I, such that supp

(

fi

)

⊂

B+,fihas

exactlyonemonomial

γ (

fi

)

in

∂

B(also called theleading monomialoffi) such that if

γ (

fi

)

=

γ (

fj

)

theni

=

j,

Remark that the elements ofFcan be seen as rewriting rules for the leading monomial using monomials of the setB. Definition 2.4. A reducing familyFof a degree

λ

∈

Λfor a setBis a set of polynomials, such thatFis a rewriting family for the setB,

∀

m

∈

∂

Bof degree at most

λ

,

∃

f

∈

F

|

γ (

f

)

=

m.

For the setB

= {

1

,

x0

,

x1

,

x0x1

}

, the set of polynomialsF

= {

x20

−

1

,

x 2

1

−

x1

,

x20x1

−

x1

,

x21x0

−

x1

}

is a reducing family of

degree 3.

A reducing family of degree

λ

for a setB(connected to 1) allows us to rewrite the monomials of

h

B+

_i

λ moduloF as elements of

h

B

i

_λ. This leads, in fact, to the definition of the linear projection

π

F, associated to a reducing family for a setB

connected to 1.

Definition 2.5. Given a reducing familyF of degree

λ

∈

Λfor a setBconnected to 1, we define the linear projection

π

F

: h

B+

i

λ

→ h

B

i

λsuch that

∀

m

∈

B_λ

, π

F

(

m

)

=

m

,

∀

m

∈

∂

B_λ

, π

F

(

m

)

=

m

−

f

;

wheref

∈

Fis the unique member ofF, such asm

=

γ (

f

)

. We extend this construction to

h

B+

_i

λby_K-linearity.

There is a parallel with the differential algebra terminology, which we want to highlight here. To a polynomial f

(

x1

, . . . ,

xn

)

∈

Rof a degree of at most

δ

, we can associate the differential equationf

(∂

1

, . . . , ∂

n

)

Φ

(

t1

, . . . ,

tn

)

=

0, where

∂

iis the derivation with respect to the variableti. It is a linear equation inΦ. The solutionΦ

(

t1

, . . . ,

tn

)

lives in the ring K

[[

t1

, . . . ,

tn

]]

of formal power series int1

, . . . ,

tn, which we can truncate in degree

λ

Λ

(

fi

)

. In the literature this set

Jλ

(

t

)

of truncated series in degree

λ

is called the space of

λ

-jets. See, e.g. [33] for more details.

Given a set of polynomialsF

⊂

_K

[

x

]

of degree at most

λ

, we consider the so-calledsolution manifoldRλof the differential systemF

(∂)(

Φ

)

=

0 inJλ

(

t

)

. (In our case, it is just a linear space.) Given

λ

0

_≺

_λ

_andΦ

_∈

_J

λ

(

t

)

, we can delete the coefficients of degrees exceeding

λ

0, and project the set of solutions ofF

(∂)(

Φ

)

=

0 projected onJλ0

(

t

)

. Then this set would be defined by the equationsF_λ0

(

Φ0

)

=

0 where

h

F_λ0

i = h

F

i ∩

_K

[

x

]

_λ0, and we denote it by

π

_λ0

(

F

)

. We denote byD

(

F

)

the new system that extendsFwith the equations

∂

if

(∂)

=

0, fori

∈

1

. . .

n,f

∈

F, obtained by formal multiplication by the

∂

i. This system

is called the prolongation ofF.

The polynomial systemFof degree at most

λ

is said to beformally integrableif

π

_λ

(

Dr+1

₍

_F

₎₎

₌

_{Ffor anyr}

_≥

_{0. A technical}

condition of involutivity is introduced to ensure the regularity of the differential system [30]. A result of Cartan–Kuranishi [4,17,30] asserts that any systemFcan be transformed by prolongation and projection into a (involutive) formally integrable

system. The connection of this construction with the notion of Mumford regularity of polynomial systems has been detailed in [21]. This correspondence has been used explicitly for solving polynomial equations in [31]. This involutive division is also used to construct a family of normal form projection related to the so-called involutive bases (see e.g. [2]). Unlike the Janet basis construction, however, the variables do not play a symmetric role. The so-called multiplicative variables are used to perform reduction and non-multiplicative ones to extend the polynomial vector space.

In the following, instead of working in the vector space of all polynomials of a given degree, we will consider completion procedures based on prolongations and projections relative to a specific set of monomialsB. Dealing locally with this set of monomials related to the number of solutions of the system, will allow us to improve significantly the linear algebra stages when we deal with this type of methods.

In the sequel, we will make heavy use of multiplication operators by one variable that we define as follows: Mi,λ

: h

B

i

λ−

→ h

B

i

_λ

b

7→

π

_F

(

xib

).

The subscript_λis redundant, as soon as we know thatFis a reducing family of degree

λ

, and we will omit this subscript in the sequel.

Definition 2.6. LetF

= {

f1

, . . . ,

fs

}

be a polynomial set, we denote byFhλithe vector space:

Fh_λi

= h{

xαfi

|

Λ

(

xαfi

)

≤

λ

}i

.

Obviously, we haveFhλi

⊂

(

F

)

λwhere

(

F

)

is the ideal generated byF. Next we introduce a definition, which weakens the

notion of monomial ordering for Gröbner basis.

Definition 2.7. We say that a function

γ

:

_K

[

x

] →

M(whereMis the set of all monomials in the variablesx1

, . . . ,

xn) is

a choice function refining the gradingΛif, for any polynomialp,

γ (

p

)

is a monomial such that (a)

γ (

p

)

∈

supp

(

p

)

, (b) if m

∈

supp

(

p

)

,m

6=

γ (

p

)

then

γ (

p

)

does not dividem, and (c)Λ

(γ (

p

))

=

max

{

Λ

(

m

),

m

∈

supp

(

p

)

}

. The coefficient of the monomial

γ (

p

)

inpwill be denoted by

κ(

p

)

.

Example 2.8. In the following, we consider aMacaulay1_{choice function}

_γ

_{, such that for allp}

_∈

K

[

x

]

, the function

γ (

p

)

=

xα1

1

· · ·

xαnn satisfies the following property

|

γ (

p

)

| =

max

{|

m

|;

m

∈

supp

(

p

)

} =

d, and

∃

i0st.

α

i0

=

max

{

degxi

(

m

),

m

∈

supp

(

p

)

and

,

|

m

| =

d

;

i

=

1

, . . . ,

n

}

. In case more than one monomial satisfies these conditions, the greatest monomial according to the lexicographic order is chosen.

The monomial returned by the choice function has the same name as theleadingmonomial of an element of a reducing family. This is intended, as we will define a reducing family on behalf of the choice functions, and in this framework the two monomials coincide. Hereafter if S

= {

p1

, . . . ,

ps

}

is a polynomial set, then we denote by

γ (

S

)

the set:

γ (

S

)

=

{

γ (

p1

) . . . γ (

ps

)

}

.

Definition 2.9. Let

γ

be a choice function refining a gradingΛ. For any polynomialsp1

,

p2

∈

K

[

x

]

, let the C-polynomial

relative to

γ

and

(

p1

,

p2

)

be C

(

p1

,

p2

)

=

lcm

(γ (

p1

), γ (

p2

))

κ(

p1

) γ (

p1

)

p1

−

lcm

(γ (

p1

), γ (

p2

))

κ(

p2

) γ (

p2

)

p2

.

Let us writeΛ

(

lcm

(γ (

p1

)

,

γ (

p2

)))

for theC-degree of

(

p1

,

p2

)

and lcm

(γ (

p1

), γ (

p2

))

for the leading monomial of the pair

(

p1

,

p2

)

.

If

γ

is a monomial ordering, then this is almost the same definition as for anS-polynomial in [6]. We, however, use a new name to emphasize that now

γ

can be not a monomial ordering (i.e. a total order compatible with monomial multiplication). As we will see in the next section, theC-polynomials express commutation conditions for theMi,λ.

3. Generalized normal form criterion

LetF

= {

f1

, . . . ,

fs

}

be a polynomial system, and letIbe the ideal generated by it. Remember thatFhλi(theK-vector space

spanned by the monomial multiples of thefi,xαfiof degrees

λ

∈

Λ) is included intoIλ. Thus, whenIλ

=

Fh_λiwe can define

a normal form moduloI, up to the degree

λ

as the projection of_K

[

x

]

_λalongFhλionto a supplementary space

h

B

i

λ. Hereafter,

we consider a setBof monomials, containing 1.

LetFbe a rewriting family and letH

= {

m

,

∃

p

∈

F

, γ (

p

)

=

m

}

be the set of their leading monomials. Then, obviously, F allows us to define the projection

π

F of the intersectionB

∪

Hon the setBalong

h

F

i

. We may, however, extend this

projection using the following extension process:

Definition 3.1. LetFbe a rewriting family. For allm

∈

B, we define

π

_Fe

(

m

)

=

m. Form

6∈

B, there existsm0

∈

∂

Bandr integersi₁

, . . . ,

i_r

∈ [

1

,

n

]

, such thatm

=

x_i1

· · ·

x_irm0_{. We define}

_π

e

F

(

m

)

by induction onk, as follows.

•

ifr

=

0,

π

e F

(

m 0

₎

_{is defined as}

_π

e F

(

m 0

₎

₌

_π

F

(

m0

)

=

m0

−

f wheref

∈

Fis such that

γ (

f

)

=

m0.

• ∀

k

,

1

≤

k

≤

r,

π

_Fe

(

xir−k

· · ·

xirm

0

₎

₌

_π

F

(

xir−k

π

e

F

(

xir−k+1

· · ·

xirm

0

₎₎

_{where the latter quantity is defined. Otherwise, we say}

that

π

_Fe

(

m

)

is undefined.

Remark that the above process allows us to define

π

_Feonly on monomials, and we extend it implicitly, by linearity. Remark also, that this extension process is not defined in a unique way. Indeed, two different decompositions of a monomialmmay lead to two different values of

π

_Fe

(

m

)

. The following theorem, however, shows that this extension process becomes canonical as soon as we check some commutativity conditions.

Theorem 3.2. Assume that B is connected to1. Let F be a rewriting family, and let E be the set of monomials m such that for all decompositions of m into the products of variables, m

=

xi0

· · ·

xik,

π

F

(

xi0

π

F

(

xi1

· · ·

π

F

(

xik

)

· · ·

))

is defined. Suppose that for all

m

∈

E and all indices i

,

j

∈ [

1

,

n

]

, such that xixjm

∈

E, we have

π

e

F

(

xi

π

Fe

(

xjm

))

=

π

Fe

(

xj

π

Fe

(

xim

)).

Then

π

e

F coincides with the linear projection

π

Sof

h

E

i

on

h

B

i

along the vector space spanned by the polynomials S

= {

xαf

, α

∈

Nn

,

f

∈

F and xα

γ (

f

)

∈

E

}

.

Proof. Remark that the way we define it, makes

π

_Feinherently alinearmultivalued map. Hence to prove the theorem, we first have to show that, under the above hypotheses,

π

_Febecomes a well defined map, and next that this well defined linear map coincides with the projection

π

Sof

h

E

i

on

h

B

i

along

h

S

i

.

Remark also thatEis obviously stable by monomial division: if all the possible decompositions ofmas a product of variablesm

=

xi0

· · ·

xikare such that

π

F

(

xi0

· · ·

π

F

(

xik

))

is defined, then a fortiori ifm

0

is a divisor ofm, this property is true form0_{. Let us show that the extension process is independent of the waymis decomposed as a product of variables. Let}

m

=

xi0m

0

₌

_x i1m

00_withi

0

6=

i1andm

,

m0

,

m00

∈

E. Then there existsm000

∈

E(sinceEis stable by monomial division), such

thatm

=

xi0xi1m

000

. Sincem

,

m0

,

m00

,

andm000are inE, the projections

π

_Fe

(

m0

)

,

π

_Fe

(

m00

)

,

π

_Fe

(

xi0m

0

₎

,

π

_Fe

(

xi1m

00

_),

and

π

_Fe

(

m000

)

are defined and we have:

π

e F

(

xi0

π

e F

(

m 0

₎₎

₌

_π

e F

(

xi0

π

e F

(

xi1

π

e F

(

m 000

_))),

π

e F

(

xi1

π

e F

(

m 00

₎₎

₌

_π

e F

(

xi1

π

e F

(

xi0

π

e F

(

m 000

_))).

The commutation condition guarantees that the two quantities are equal, so that the definition of

π

e

F does not depend on

the way of writingmas a product of variables.

Next, we have to show that the projections

π

_Feand

π

Scoincide on their common set of definition. We do this by induction

on the size of the monomials as follows:

It is true that

π

_Fe

(

1

)

=

π

_S

(

1

)

=

1 (since 1

∈

B). For any monomialm

6=

1 inE, the property of connectivity ofBand the definition ofEimplies that

∃

m0

∈

Eandi0

∈ [

1

,

n

]

such thatm

=

xi0m

0

and

π

_Fe

(

m0

)

is defined, so that we have

π

e F

(

m

)

=

π

Fe

(

xi0m 0

₎

₌

def

_π

e F

(

xi0

π

e F

(

m 0

₎₎

₌

induction

_π

e F

(

xi0

π

S

(

m 0

₎₎

_{∈ h}

B

i

.

Now by induction,m0

₋

_π

S

(

m0

)

∈

Sλ−where

λ

=

Λ

(

m

)

and m

−

π

_Fe

(

m

)

=

xi0

(

m 0

₋

_π

S

(

m0

))

+

xi0

π

S

(

m 0

₎

₋

_π

e F

(

xi0

π

S

(

m 0

₎₎

∈ h

S

i

.

Thus

π

_Fe

(

m

)

is the projection ofmon

h

B

i

along

h

S

i

.

Now, suppose that we are given a reducing family of degree

δ

instead of a rewriting family. Then, we can further extend the above theorem with the help from the following lemma.

Lemma 3.3. Let F be a reducing family of degree

λ

, for a set B connected to1and suppose that

∀

f

∈

F

,

Λ

(γ (

f

))

=

Λ

(

f

)

. With the notation ofTheorem3.2, the set E contains the set of monomials of degree less than or equal to

λ

.

Proof. Letm

∈

Mλ be a monomial of degree less than or equal to

λ

, so thatmcan be written asm

=

xi1

. . .

xid with

d

= |

m

|

. Let us prove by induction onk

≤

dthat the projection pk

=

π

F

(

xik

π

F

(

· · ·

π

F

(

xi1

))

· · ·

)

is defined and that Λ

(

pk

)

≤

Λ

(

xik

· · ·

xi1

)

.

Considerxik+1pk

∈

B

+_{. Form}0

_∈

_supp

₍

_x

ik+1pk

)

∩

B, we have

π(

m

0

₎

₌

_m0_{and by the induction hypothesisΛ}

₍

_m0

₎

_≤

Λ

(

xik+1

· · ·

xi1

)

. We have a rewriting rule for m

0

_∈

supp

(

xik+1pk

)

∩

∂

BbecauseF is a reducing family of degree

λ

. The hypothesis that

∀

f

∈

F

,

Λ

(γ (

f

))

=

Λ

(

f

)

implies thatm0_{rewrites in terms of the monomials of degrees bounded by}

Λ

(

xik+1

· · ·

xi1

)

. Thereforepk+1

:=

π

F

(

xik+1pk

)

is defined andΛ

(

pk+1

)

≤

Λ

(

xik+1

· · ·

xi1

)

. By induction

π

_Fe

(

xi1

· · ·

π

e

F

(

xid

))

is defined for any decompositionm

=

xi1

· · ·

xid

∈

Mλ, so thatm

∈

E. This completes the

proof.

Theorem 3.4. Let F be a reducing family of degree

λ

for a set B connected to1. If we have:

• ∀

f

∈

F

,

Λ

(γ (

f

))

=

Λ

(

f

)

.

•

Mj,λ

◦

Mi,λ−

=

M_i,λ

◦

M_j,λ−, for1

≤

i

,

j

≤

n,

Proof. ByLemma 3.3, we haveE

⊃

MλbecauseFis a reducing family of degree

λ

.

Let us prove that for allm

∈

Mλ−− and all pairs of indices

(

i

,

j

)

, there exists a way to define

π

_Fe such that

π

_Fe

(

x_i

π

e

F

(

xjm

))

=

π

Fe

(

xj

π

Fe

(

xim

)).

AsM_λ−−

⊂

M_λ

⊂

E,

π

_Fe

(

m

)

is defined and supp

(π

_Fe

(

m

))

⊂

B. We define

π

_Fe

(

x_im

)

=

π

_F

(

x_i

π

_Fe

(

m

))

and, similarly,

π

e

F

(

xjm

)

=

π

F

(

xj

π

Fe

(

m

))

. With this definition we have:

π

e

F

(

xi

π

Fe

(

xjm

))

=

Mi,λ

(

Mj,λ−

(π

e

F

(

m

)))

=

Mj,λ

(

Mi,λ−

(π

_Fe

(

m

)))

=

π

_Fe

(

x_j

π

_Fe

(

x_im

)),

which proves the commutation property. We end the proof by applyingTheorem 3.2. Corrolary 3.5. With the hypothesis ofTheorem3.4, we haveK

[

x

]

λ

= h

B

i

_λ

⊕

Fhλi.

Let us show another effective way of checking whether we can define a projection fromFhλi(vector space spanned by the

monomial multiples of thefiof degree

λ

) onto

h

B

i

λ(the elements of degree

λ

in the vector space spanned byB) starting from

a reducing family of degree

λ

, and without computingexplicitlythe multiplication operators.

Theorem 3.6. Let

λ

∈

Λ. Let F be a reducing family of degree

λ

∈

Λ, for B. Assume that

∀

f

∈

F

,

Λ

(γ (

f

))

=

Λ

(

f

)

and let

π

F

be the induced projection from

h

B+

_i

λonto

h

B

i

_λ. Then

∀

f

,

f0

_∈

_F

hλisuch that C

(

f

,

f0

)

∈ h

B+

i

λ,

π

F

(

C

(

f

,

f 0

₎₎

₌

0

iff

π

Fextends uniquely as a projection

π

FefromK

[

x

]

λonto

h

B

i

λsuch thatker

(π

e F

)

=

Fhλi.

Proof. ByTheorem 3.4, we have to show that this condition is equivalent to the commutation of the operatorsMi,λ0

, λ

0

< λ

on the monomials ofB_λ−−.

For anym

∈

B_λ−− and anyi₁

6=

i₂, such thatx_i

1m

∈

∂

B,xi2m

∈

∂

B, there existsf

,

f 0

_∈

_F h_λ−_isuch that

γ (

f

)

=

x_i 1m,

γ (

f0

₎

₌

_x i2m. Thus, we have

π

F

(

xi1m

)

=

γ (

f

)

−

f,

π

F

(

xi2m

)

=

γ (

f 0

₎

₋

_f0_andC

₍

_f

_,

_f0

₎

₌

_x i2f

−

xi1f 0

_{∈ h}

_B

_i

+ λ. Consequently, Mi2,λ

(

Mi1,λ−

(

m

))

−

Mi1,λ

(

Mi2,λ−

(

m

))

=

Mi2,λ

(γ (

f

)

−

f

)

−

Mi1,λ

(γ (

f 0

₎

₋

f0

)

=

π

_F

(

xi2

γ (

f

)

−

xi2f

)

−

π

F

(

xi1

γ (

f 0

₎

₋

xi1f 0

₎

=

π

_F

(

xi1f 0

₋

xi2f

)

=

π

F

(

C

(

f 0

_,

f

)),

which is zero by hypothesis. A similar proof applies ifxi1m

∈

Borxi2m

∈

B.

Conversely, since ker

(π

_Fe

)

=

FhλiandC

(

f

,

f0

)

∈

Fhλi

∩ h

B

i

+_λ, we have that

π

F

(

C

(

f0

,

f

))

=

π

Fe

(

C

(

f 0

_,

f

))

=

0, which proves the equivalence andTheorem 3.6.

Remark 3.7. In this proof, we have shown that if theC-polynomials up to the degree

λ

reduce to 0, then the multiplication operatorsMi,λcommute.

Definition 3.8. A reducing familyFfor all degrees

λ

∈

Λon a setBof monomials, connected to 1 such that

π

F extends

uniquely toRwill be called aborder basisforB.

Finally, the previous results lead us to a new proof of Theorem 3.1 of [25]:

Theorem 3.9. Let F be a reducing family for a set B of monomials, connected to1, let

π

F be the corresponding reduction from

h

B+

_i

_onto

_h

_B

_i

_{, and let M}

i

: h

B

i → h

B

i

, such that

∀

b

∈ h

B

i

, Mi

(

b

)

=

π

F

(

xib

)

. Then

Mj

◦

Mi

=

Mi

◦

Mj

,

for 1

≤

i

,

j

≤

n

iff there exists a unique projection

π

_FefromK

[

x

]

onto

h

B

i

such thatker

(π

Fe

)

=

(

F

)

and

(π

e

F

)

|hB+_i

=

π

_F.

Proof. Under the above hypotheses, byTheorem 3.4, the projection

(π

F

)

|hB+_iλextends uniquely for any

λ

∈

Λto a projection

π

e

FλfromK

[

x

]

λonto

h

B

i

λ, such that ker

(π

e

Fλ

)

=

Fhλi. Since for any

λ, λ

0

∈

Λsuch that

λ

≺

λ

0, we have

(

Bλ0

)

_λ

=

B_λ, and Fhλi

⊂

(

Fhλ0_i

)

_λ, we also have

(π

e

F_λ0

)

|K[x]λ

=

π

Feλ. This defines a unique linear operator

π

e F onK

[

x

]

, such that

π

e |K[x]λ

=

π

e Fλ and ker

(π

e F

)

=

P

λ∈_ΛFhλi

=

(

F

)

. We have completed the proof of the direct implication. The converse implication is

immediate.

4. Syzygies and commutation relations

In this section, we analyze more precisely the relations between the polynomials of the border basisF

=

(

f_ω

)

_ω∈∂Bwhere

f_ω

=

ω

−

ρ

_ωwith

ρ

_ω

∈ h

B

i

. These relations or syzygies form a module that we denote by Syz

(

F

)

=

(

X

ω h_ωe_ω

∈

_K

[

x

]

∂B

;

X

ω h_ωf_ω

=

0

)

,

where

(

e_ω

)

_ω∈∂Bis the canonical basis ofK

[

x

]

∂B. IfFis a border basis family constructed from an initial set of polynomials

H

= {

h1

, . . . ,

hs

}

, we can expressfω

∈

Fin terms of the polynomials inH(and conversely) so that a syzygy onFinduces a

syzygy onH(and conversely). Therefore, we are going to consider only the syzygies onF. For anyb

∈ h

B

i

andi

∈ [

1

,

n

]

,xib

∈ h

B+

i

can be projected in

h

B

i

alongF:

π

F

(

xib

)

=

xib

−

X

ω∈∂B

µ

i b,ωfω

.

More generally, for anyp

∈

_K

[

x

]

, we denote by

µ

i_p,ω the coefficient

µ

i_b,ωin

π

F

(

xib

)

forb

=

π

F

(

p

)

∈ h

B

i

and write

µ

i

₍

_p

₎

₌

P

ω∈∂B

µ

i

pfω. Notice that ifxip

∈

Bthen

µ

i

(

p

)

=

0.

For anym

=

xi3

· · ·

xik

∈

Bandi1

,

i2

∈

1

. . .

n, the two decompositions

π

F

(

xi1

π

F

(

xi2m

))

=

π

F

(

xi2

π

F

(

xi1m

))

yield the syzygy xi1

X

ω∈∂B

µ

i2 m,ωfω

−

xi2

X

ω∈∂B

µ

i1 m,ωfω

−

X

ω∈∂B

(µ

i2 xi₁m,ω

−

µ

i1 xi₂m,ω

)

fω

=

0

.

(1)

These relations can be also rewritten as: xi1

µ

i2

(

_m

)

₊

µ

i1

(

_x

i2m

)

−

xi2

µ

i1

(

_m

)

₋

µ

i2

(

_x

i1m

)

=

0

.

We denote byΞ the module ofK

[

x

]

∂B generated by these syzygies. It is also the module generated by the relations of

commutationMi1

◦

Mi2

(

b

)

−

Mi2

◦

Mi1

(

b

)

=

0 for allb

∈

B,ii

,

i2

∈ [

1

. . .

n

]

. We distinguish the following relations:

•

Ifxi1m

,

xi2m

∈

B, then

µ

i1

(

_m

)

₌

µ

i2

(

_m

)

₌

_0,

µ

i1

(

_x

i2m

)

=

µ

i2

(

_x

i−1m

)

=

0 andmdoes not yield a non-trivial relation of

the form (1).

•

Ifxi1m

∈

Bbutxi2m

∈

∂

B, andxi1xi2m

∈

∂

B, then

π

F

(

xi2m

)

=

xi2m

−

fxi₂mand we have the relation

xi1fxi2m

−

fxi1xi2m

+

X

ω∈∂B

µ

i1

xi₂m,ωfω

=

0

.

(2)

•

Ifxi1m

∈

B,xi2m

∈

∂

B, andxi1xi2m

∈

B, then

π

F

(

xi2m

)

=

xi2m

−

fxi₂mand we have the relation

xi1fxi₂m

+

X

ω∈∂B

µ

i1

xi₂m,ωfω

=

0

.

(3)

•

Ifxi1m

∈

∂

Bandxi2m

∈

∂

B, then

π

F

(

xi1m

)

=

xi1m

−

fxi1m,

π

F

(

xi2m

)

=

xi2m

−

fxi2mand we have the syzygy xi1fxi₂m

−

xi2fxi₁m

−

X

ω∈∂B

(µ

i2 xi1m,ω

−

µ

i1 xi2m,ω

)

fω

=

0

.

(4)

The syzygies (2) and (4) are called respectivelynext-doorandacross-the-street relationsin [16]. The syzygies (3) do not exist ifBis stable under division (since in this case,xi2m

6∈

Bimpliesxi1xi2m

6∈

B). All these syzygies are simply the non-trivial relations induced by the ‘‘border’’C-polynomials (seeDefinition 2.9).

Now let us prove that the module of syzygies is generated by these commutations syzygies. For any monomialm

=

xi1

· · ·

xik, we can rewrite by induction its projection as:

π

F

(

m

)

=

π

F

(

xi1

π

F

(

xi2

· · ·

)

=

m

−

X

l=1...k xi1

· · ·

xil−1

X

ω∈∂B

µ

il x_il+1···xk,ωfω (5)

with the convention thatxik+1

=

1. We denote by Ξxi1,...,xik

=

X

l=1...k xi1

· · ·

xil−1

X

ω∈_∂B

µ

il x_il+1···xk,ωeω

the corresponding element ofK

[

x

]

∂B. Notice that this decomposition depends on the order of the decompositionm

=

xi1

· · ·

xikas a product of variables.

Lemma 4.1. If m

=

xi1

· · ·

xik

=

xj1

· · ·

xjkthen Ξxi₁···x_ik

−

Ξxj₁,...,x_jk

∈

Ξ

.

Proof. First consider a permutation of two variablesm

=

m1xilxil+1m2

=

m1xil+1xilm2 (withm1

=

xi1

· · ·

xil−1,m2

=

xil+2

· · ·

xik). Using (5), the two ways of projectingm

=

m1xilxil+1m2

=

m1xil+1xilm2yield the syzygy

Ξ...,x_il,x_il+1,...

−

Ξ...,x_il+1,x_il,...

=

m1 xil

X

ω∈_∂B

µ

il+1 m2,ωeω

−

xil+1

X

ω∈_∂B

µ

il m2,ωeω

+

X

ω∈_∂B

(µ

il x_il+1m2,ω

−

µ

il+1 x_ilm2,ω

)

eω

!

which is inΞ. By iterated permutations of two variables, we transform the sequence xi1

, . . . ,

xik into the sequence

Lemma 4.2. If m

θ

=

m0

θ

0with

θ, θ

0

∈

∂

B and m

,

m0

∈

M, then m e_θ

≡

m0e_θ0

+

X

ω∈∂B

p_ωe_ωmodΞ

with p_ω

∈

_K

[

x

]

of degree

<

max

(

|

m

|

,

|

m0

|

)

.

Proof. Ifm

=

xi1

· · ·

xidwith

θ

=

xid+1b

∈

∂

Bandb

∈

B, the projection formula (5) ofm

θ

has the form

π

F

(

m

θ)

=

m

θ

−

xi1

· · ·

xidfθ

−

X

l=1...d xi1

· · ·

xil−1

X

ω∈∂B

µ

il x_il+1···x_idθ,ωfω sincexid+1b

=

θ

and

π

F

(θ)

=

θ

−

fθ. Ifm 0

₌

xj1

· · ·

xjd0 and

θ

0

_∈

_∂

Bwithm

θ

=

m0

θ

0, then byLemma 4.1the two decompositions yield the syzygy

xi1

· · ·

xideθ

−

xj1

· · ·

xjd0eθ0

+

X

l=1...d xi1

· · ·

xil−1

X

ω∈∂B

µ

il x_il+1···x_idθ,ωeω

−

X

l=1...d0 xj1

· · ·

xjl−1

X

ω∈∂B

µ

jl x_jl+1···xj_d0θ0,ωeω as an element ofΞ. The syzygy is of the formm e_θ

−

m0_e

θ0

+

P

_ω

∈∂Bpωeωwith deg

(

pω

) <

max

(

d

,

d

0

₎

_{, which proves the}

lemma.

This yields the following result for a border basis, conjectured in [16] for the case whereBis stable by division: Theorem 4.3. Let B

⊂

Mbe connected to1and let F

=

(

f_ω

)

_ω∈_∂Bbe a border basis for B. Then Syz

(

F

)

is generated by the relations

(2),(3)and(4). Proof. Let

σ

=

P

ωpωe_ω

∈

Syz

(

F

)

. In this sum consider any term

λ

m e_θ where

λ

∈

_K

− {

0

}

,m

∈

M,

θ

∈

∂

Band

δ

B

(

m

θ) <

|

m

| +

1. Then there existsm0

∈

M,

θ

0

∈

∂

Bsuch thatm

θ

=

m0

θ

0and

δ

B

(

m0

θ

0

)

= |

m0

| +

1. This implies that

|

m

|

>

|

m0

_|

_{. ApplyLemma 4.2to reduce the productm e}

θ moduloΞ, to m0e_θ0

+

X

ω∈∂B

q_ωe_ω

,

with deg

(

q_ω

) <

|

m

|

. Iterate this reduction, assuming for each term

λ

m e_ω(

λ

∈

_K

− {

0

}

,m

∈

supp

(

p_ω

)

,

ω

∈

∂

B) that

δ

B

(

m

ω)

= |

m

| +

1. Notice that the polynomialm fωhas only one monomial of maximalB-index, which ism

ω

.

Since

P

ωpωf_ω

=

0, there exist

θ

6=

θ

0

∈

∂

Band monomialsm

∈

supp

(

p_θ

)

,m0

∈

supp

(

p_θ0

)

such thatm

θ

=

m0

θ

0and

δ

B

(

m

θ)

is maximal among all terms of the syzygy. ByLemma 4.2, we can replace this pair

(

m eθ

,

m0eθ0

)

moduloΞby a sum of terms of smaller degree. This transformation reduces either the number of terms with maximalB-index or the degree of the polynomialsp_ω.

Since we cannot iterate it infinitely, we deduce that

σ

is inΞ. 5. Algorithmic issues

Based on our study in the previous sections, we can arrive at an algorithm in [29]. The main idea is to translate the previous concepts into linear algebra setting. According to Section3, to compute effectively a normal form, one has to find a (monomial) basisBof the quotient algebra connected to 1 and to define border relations such that the multiplication operators commute. The algorithm in [29] is a fix point method, which updates

•

a potential monomial basisBof the quotient algebra and

•

a setPof polynomials or rewriting rules (with one monomial of their support in

∂

Band the remaining monomials inB), until a fix point is reached. At each step of the algorithm, the following operations are performed:

1. The setP0_{of polynomials ofP}+_{with support inB}+_{is computed.}

2. By taking linear combinations, a basisP

˜

of the vector space

h

P0

i

is computed such that each element of this basis has at most one monomial in

∂

B(and has the other monomials inB).

3. TheC-polynomials of the elements ofPwith their support inB+_{are computed and reduced byP.}

˜

4. If non-zero polynomials with support inBappear, the potential basisBand the polynomial setPare updated. The update ofBis done by removing some parts ofB. The update ofPis done by combining the elements ofP

˜

and the reduced C-polynomials, in order to obtain rewriting rules for the new setB.

The details and the technical proof of termination of the algorithm are available in [29]. We mention here that we have improved the algorithm since then: unlike [29] we avoid repeated execution of similar reductions, when we treat the degree drops.

6. Stability of the bases

This section is devoted to the study of the stability under numerical perturbations of the bases computed by the previous algorithm.

In many real-life problems, the systemf

=

(

f1

, . . . ,

fs

)

to be solved is given only with limited accuracy. However, most of

the time, one also knows that the structure of the solutions is invariant in a small neighborhood of the system. Thus, it is often required that the polynomial solvers produce a representation of the quotient algebra that is stable in a small neighborhood of the initial system. The structural numerical stability of the basis is required in order to have a smooth behavior of the coefficients of the representation in the neighborhood off

=

(

f1

, . . . ,

fs

)

. By definition, a neighborhood offis an open set

(containingf) in the vector space of polynomials

(

h1

, . . . ,

hs

)

such thatΛ

(

hi

)

Λ

(

fi

)

(i

=

1

, . . . ,

n). For

>

0, we define

byN

(

f

)

the set of systems

(

h1

, . . . ,

hs

)

such thatΛ

(

hi

)

Λ

(

fi

)

and the coefficient vector ofhiis at most at distance

from

the coefficient vector offi(for the

∞

-norm).

Assumption 6.1. Hereafter

γ

denotes a choice function refining a reducing gradingΛsuch that, for allp

∈

R,

γ (

p

)

depends only on the support ofpand not on the numerical value of its monomial coefficients (e.g., Macaulay’s choice function, grevlex choice function,. . . ). Let

γ

be the choice function that for anyp

∈

R, applies the choice function

γ

on the monomials ofp, whose coefficient norm is larger than

.

Theorem 6.2. Letf

=

(

f1

, . . . ,

fs

)

be a zero dimensional polynomial system, such that in a neighborhood U off, all systems have

the same number D of complex solutions, counted with multiplicities. Let a positive

be small enough, there exists

ν >

0such that for any systemf0

_∈

_N

ν

(

f

)

⊂

U , the basis B computed with

γ

satisfyingAssumption6.1for the systemfis also a basis for the systemf0.

Proof. Consider the matrixM, whose rows correspond to the coefficient vectors of the monomial multiplesm fi, with

deg

(

m

)

+

deg

(

fi

)

≤

κ

+

d0where

κ

is the number of loops in Algorithm [29] andd0is the maximum degree of the polynomials

fi. The columns are indexed by all the monomials of degree

≤

κ

+

d0.

We denote byBthe monomial set obtained as a basis ofA

=

R

/

Iby applying Algorithm [29] to the polynomialsf1

, . . . ,

fs

with the choice function

γ

. By construction, the set of monomials indexing the columns ofMcontains

∂

B. LetMg_{be the same matrix asM, but constructed with the polynomialsf}

ireplaced bygenericequations, i.e. equations with

indeterminate coefficients having the same support asf1

, . . . ,

fs. LetNbe the block of columns ofMindexed by monomials

not inBand letNgbe the corresponding block inMg.

Since the operations of the algorithm in [29] consist in computing linear combinations of some monomial multiples of the polynomialsP(see Section5and [29]), and thus of monomial multiples offiof degree

≤

κ

+

d0, the complete computation

can be reinterpreted as an optimized triangulation procedure of the blockN.

The blockNgspecialized atfis invertible, because any monomial not inBcan be reduced by the computed border basis offto an element in

h

B

i

. This implies thatNg_{specialized atf}0_{is also invertible, forf}0

_∈

_N

ν

(

f

)

and sufficiently small positive

ν

. Therefore, any monomial of

∂

Bcan be reduced modulo the ideal

(

f0

₎

_{to an element in}

_h

_B

_i

_{. Consequently,B(which contains}

1) is a generating set of the quotient algebraA0

=

R

/(

f0

₎

for anyf0

_∈

N_ν

(

f

)

.

Since the number of solutionsD

= |

B

|

(counted with multiplicity) is left unchanged by small perturbations in the neighborhood N_ν

(

f

)

of f, the monomial setB, which has cardinality exactly D, is also a basis of the quotient algebra A0

=

R

/(

f0

)

for the perturbed systemf0.

Consider now a slight modification of the algorithm in [29]: the coefficients whose norm is less than

are simply ignored in all the steps of the algorithm. This means that they will not be taken into account when we choose aleadingmonomial, or deciding if a polynomial is nonzero. This minor variation of the algorithm will be called the

-algorithm in the next theorem. Similar deletion of the absolutely small input coefficients is routine in numerical algorithms. Remark, however, that the following theorem is stated rigorously and that the computed result isnotan approximation of the true quotient algebra!

Theorem 6.3. Letf

=

(

f1

, . . . ,

fs

)

be a zero dimensional polynomial system, such that in a neighborhood U offall systems have

the same number D of complex solutions, counted with multiplicities. Let a positive

be small enough. Then there exists

ν >

0, such that for any systemf0

∈

N_ν

(

f

)

⊂

U , the basis B computed with

γ

satisfyingAssumption6.1for the systemfis also the basis obtained with

γ

and the

-algorithm for the systemf0

.

Proof. ByTheorem 6.2,Bis also a basis of the quotient algebraR

/(

f0

₎

. This basisBis obtained by applying Algorithm [29] to the systemf. The result of this algorithm does not change if we replace the choice function

γ

by

γ

where a positive

is small enough (e.g., is smaller than the minimum of the norm of the coefficients of the polynomials on which

γ

is applied).

Let us show by induction on the loop indexkof the algorithm, that the steps of and the polynomials computed by the

-algorithm with

γ

are the same as for the direct algorithm, up to the terms of norm smaller that

.

This is clearly true for the first stepk

=

1. Now, suppose that steps 1

, . . . ,

k0_{of Algorithm [29] ran onfand its}

_variant