On the Uniqueness of Simultaneous Rational Function Reconstruction
aa r X i v : . [ c s . S C ] F e b On the Uniqueness of Simultaneous Rational FunctionReconstruction
Eleonora Guerrini, Romain Lebreton, Ilaria Zappatore guerrini,lebreton,[email protected], Université de Montpellier, CNRSMontpellier, France
ABSTRACT
This paper focuses on the problem of reconstructing a vector of ra-tional functions given some evaluations, or more generally giventheir remainders modulo different polynomials. The special caseof rational functions sharing the same denominator, a.k.a.
Simul-taneous Rational Function Reconstruction (SRFR), has many appli-cations from linear system solving to coding theory, provided thatSRFR has a unique solution. The number of unknowns in SRFRis smaller than for a general vector of rational function. This al-lows to reduce the number of evaluation points needed to guaran-tee the existence of a solution, but we may lose its uniqueness. Inthis work, we prove that uniqueness is guaranteed for a genericinstance.
The Vector rational function reconstruction (VRFR) is the problemof finding all rational functions v / d = ( v / d , . . . , v n / d n ) whichsatisfy some degree constraints, given a certain number of theirevaluations ( v / d )( α j ) = ω j . We consider a generalized version ofthis problem, where we suppose to know the images modulo dif-ferent polynomials a , . . . , a n , i.e. u i = v i / d mod a i for 1 ≤ i ≤ n .The Simultaneous Rational Function Reconstruction (SRFR) problemis a particular case of the vector rational function reconstructionwhere the rational functions v / d = ( v / d , . . . , v n / d ) share thesame denominator (see Section 2.1). We can apply the SRFR indifferent problems: from the decoding of classic and interleavedReed-Solomon codes to the polynomial linear system solving. Asin the classic rational function reconstruction we focus on the ho-mogeneous linear system related to our equations in its weakerform, i.e. v − d u ≡ a . If the number of equations is equalto the number of unknowns minus one then there always existsa non-trivial solution. From now on, we will assume to be in thiscase. Note that the common denominator constraint of SRFR im-plies less unknowns than general VRFR, so less equations. Thishas a direct impact on the complexity of its applications. However,the uniqueness in not anymore guarantied as shown in Counterex-ample 2.2. Having a unique solution is fundamental for decodingalgorithms or Evaluation-Interpolation methods (like for instancein linear system solving). This paper focuses on the conditions thatguarantee the uniqueness of solutions of the SRFR.Previous works show that in the application of SRFR for polyno-mial linear system solving, the uniqueness is ensured under somespecific degree conditions [OS07]. We have reasons to believe thatwe can generalize this result: we conjecture that for almost all ( v , d ) the SRFR problem admits a unique solution (see Conjecture 2.5).We can learn more about the conditions of uniqueness fromthe results coming from error correcting codes. Interleaved Reed Solomon Codes (IRS) can be seen as the evaluation of a vector ofpolynomials v . The problem of decoding IRS codes consists in thereconstruction of the vector of polynomials v by its evaluations,some possibly erroneous. A classic approach to decode IRS codesis the application of the SRFR for instances u = v + e where e are the errors. Results from coding theory show that for all v andalmost all errors e , we get the uniqueness of SRFR for the corre-sponding instance u (provided that there are not too many errors)[BKY03, BMS04, SSB09]. There is a natural generalization of SRFRwhen errors occur (SRFRwE, see Section 2.2), which can be seenas fractional generalization of IRS [GLZ19, GLZ20]. We conjecturethat we can decode almost all codeword ( v / d ) and almost all er-rors e of this fractional code (Conjecture 2.9). In this paper wepresent a result which is a step towards this conjecture. We provethat uniqueness is guaranteed for a generic instance u of SRFR,(Theorem 5.2). Our result is valid not only given evaluations, butalso in the general context of any moduli a .Our approach to prove Theorem 5.2 is to study the degrees ofa relation module. Solutions of SRFR are related to generators ofa row reduced basis of this K [ x ] -module which have a negativeshifted-row degree. Shifts are necessary to integrate degree con-straints. We show that for generic instances, there is only one gen-erator with negative row degree, hence uniqueness of the SRFRsolution.Previous works studied generic degrees of different but relatedmodules: e.g. for the module of generating polynomials of a scalarmatrix sequence [Vil97], for the kernel module of a polynomialmatrix and specific matrix dimensions [JV05]. Both cases does notconsider any shift. The generic degrees also appear in dimensionsof blocks in a shifted Hessenberg form. However, the link with thedegree of a module is unclear and no shift is discussed (shiftedHessenberg is not related to our shift) [PS07]. We prove our resultfor any shift and any matrix dimension by adapting some of theirtechniques, and by proving that they apply to the specific relationmodules related to SRFR.In Section 2 we introduce the motivations of our work, startedfrom the classic SRFR to the extended version with errors. Wealso show their respective applications in polynomial linear sys-tem solving and in error correcting algorithms. In Section 3, wedefine the algebraic tools that we will use to prove our technicalresults of the Section 4. In Section 5 we explain how these resultsare linked to the uniqueness of the solution of the SRFR and wefinally prove the Theorem 5.2 about the generic uniqueness. leonora Guerrini, Romain Lebreton, Ilaria Zappatore In this section we recall standard definitions and we state our prob-lem, starting from rational function reconstruction and its applica-tion to linear algebra. Let K be a field, a , u ∈ K [ x ] with deg ( u ) < deg ( a ) . The Rational Function Reconstruction (shortly RFR) is theproblem of reconstructing a rational function v / d ∈ K ( x ) suchthat gcd ( d , a ) = , vd ≡ u mod a , deg ( v ) < N , deg ( d ) < D . (1)We focus on the weaker equation: v ≡ du mod a , deg ( v ) < N , deg ( d ) < D . (2)The RFR problem generalizes many problems including the Padéapproximation if a = x f and the Cauchy interpolation if a = Î fi = ( x − α i ) , where the α i are pairwise distinct elements of the field K . Thehomogeneous linear system related to the Equation (2) has deg ( a ) equations and N + D unknowns. If deg ( a ) = N + D −
1, the dimen-sion of the solution space of Eq. (2) is at least 1 and it always admitsa non-trivial solution. Moreover, such a solution is unique in thesense that all solutions are polynomial multiples of a unique one, ( v min , d min ) (see e.g. [GG13, Theorem 5.16]). On the other hand,Equation (1) does not always have a solution, but when a solutionexists, it is unique. Indeed, it is v min / d min and we can reconstruct itby the Extended Euclidean Algorithm (EEA). Throughout this paper,we will focus on Equation (2).The RFR can be naturally extended to the vector case as fol-lows. Let a , . . . , a n ∈ K [ x ] with degrees f i = deg ( a i ) and u = ( u , . . . , u n ) ∈ K [ x ] n where deg ( u i ) < f i . Let 0 < N i , D i < f i .The Vector Rational Function Reconstruction (VRFR) is the problemof reconstructing ( v i , d i ) for 1 ≤ i ≤ n such that v i ≡ d i u i mod a i , deg ( v i ) < N i , deg ( d i ) < D i . We can apply the RFR component-wise and so, if f i = N i + D i −
1, we can uniquely reconstruct thesolution.
Definition 2.1. (SRFR) Given u = ( u , . . . , u n ) ∈ K [ x ] n wheredeg ( u i ) < f i , and degree bounds 0 < N i < f i and 0 < D < max ≤ i ≤ n { f i } , we want to reconstruct the tuple ( v , d ) = ( v , . . . , v n , d ) such that v i ≡ du i mod a i , deg ( v i ) < N i , deg ( d ) < D . (3)We denote S u the set of solutions.The SRFR is then the problem of reconstructing a vector of ra-tional functions with the same denominator. Therefore, if f i = N i + D − ≤ i ≤ n , we can uniquely reconstruct the so-lution. In this case, the common denominator property allows toreduce the number of unknowns, with an impact on the degree ofthe a i ’s. In detail, the number of equations of (3) is Í ni = f i , whilethe number of the unknowns, i.e. the coefficients of v and d , is Í ni = N i + D . If n Õ i = f i = n Õ i = N i + D − Counterexample 2.2.
Let K = F , n = N = N = D = a = a = Î i = ( x − i ) = x + x + x +
2. Let u = v / d with v = ( x + , x + ) and d = x + x + a i . Then theSRFR with instance u has two K [ x ] -linearly independent solutions ( d , v ) = ( x + x + , , ) and ( d ′ , v ′ ) = ( x + , x + , x + ) .Uniqueness is a central property for the applications of SRFR:unique decoding algorithms are essential in error correcting codes,and it is also a necessary condition to use evaluation interpolationtechniques in computer algebra. The study of the bound on thenumber of equations which guaranties the uniqueness of SRFR hasalso repercussion on the complexity. Indeed, the complexity of de-coding algorithms or evaluation interpolation techniques dependson this number of equations. So decreasing this number has a di-rect impact on the complexity.We denote by s the rank of the K [ x ] -module spanned by thesolutions S u . Therefore, all solutions can be written as a linearcombination Í si = c i p i of s polynomials p i with polynomial coeffi-cients c i . The case s = a = . . . = a n = a and N = . . . = N n = N . They provedthe following, Theorem 2.3. [OS07, Theorem 4.2] Let k be minimal such that deg ( a ) ≥ N + ( D − )/ k , then the rank s of the solution space S u satisfies s ≤ k . Note that if k =
1, the solution is always unique ( s = ( a ) of VRFR. On theother hand, if k = n and deg ( a ) ≥ N + ( D − )/ n then s ≤ n which isalways true. Hence in this case the theorem does not provide anynew information about the solution space. This theorem representsa connection between the classic bound on the deg ( a ) = N + D − ideal one, i.e. deg ( a ) = N + ( D − )/ n (see Equation (4)), which exploits the common denom-inator property. They also proposed an algorithm that computesa complete basis of the solution space using O( nk ω − B ( deg ( a ))) operations in K where 2 ≤ ω ≤ B ( t ) : = M ( t ) log t where M is the classicpolynomial multiplication arithmetic complexity (see [GG13] forinstance). In [RS16] the complexity was improved. In particular,they introduced an algorithm that computes the solution space (inthe general case of different moduli, i.e. a , . . . , a n ) with complex-ity O( n ω − B ( f ) log ( f / n ) ) where f = max ≤ i ≤ n { deg ( a i )} .We now came back to general case of the SRFR. The main resultof this work is to prove that when the degree constraints guaranteethe existence of the solution, then for almost all u we also get theuniqueness (see Theorem 5.2). Theorem 2.4.
If Equation (4) is satisfied, then for almost all in-stances u the SRFR admits a unique solution, i.e. it has rank s = . We will both use the expressions “almost all” or “generic”, mean-ing that there exists a polynomial R such that a certain propertyis true for all instances that do not cancel R . In our case, we statethat there exists a polynomial R such that the SRFR admits a uniquesolution for all instances u such that R ( u ) , Application to polynomial linear system solving.
Suppose that wewant to compute the solution of a full rank polynomial linear sys-tem, y ( x ) = A − b ∈ K ( x ) where A ∈ K [ x ] n × n and b ∈ K [ x ] n × , n the Uniqueness of Simultaneous Rational Function Reconstruction from its image modulo a polynomial a ( x ) . We will refer to thisproblem as polynomial linear system solving (shortly PLS). We re-mark that, by the Cramer’s rule, y is vector of rational functionswith the same denominator: PLS is then a special case of SRFR. In[OS07], the authors proved that the solution space is uniquely gen-erated ( s =
1) when deg ( a ) ≥ N + ( D − )/ n in the special case of D = N = n deg ( A ) and deg ( A ) = deg ( b ) . They exploited anotherbound on the degree of a based on [Cab71].In view of Theorem 2.4 and as our experiments suggest, wecould hope for the following, Conjecture 2.5.
If Equation (4) is satisfied then for almost all ( v , d ) with gcd ( d , a i ) = , the SRFR with u = v d as input admits aunique solution. Since we have proved the uniqueness for generic instances u , itwould be sufficient to show the existence of an instance u of theform v / d to prove the conjecture. In this section we introduce the problem of the Simultaneous Ratio-nal Function with Errors ([BK14, KPSW17, GLZ19, Per14, GLZ20]), i.e. the SRFR in a scenario where errors may occur in some evalua-tions. Throughout this section we suppose that K is a finite field ofcardinality q , we fix α = { α , . . . , α f } pairwise distinct evaluationpoints in K and we consider the polynomial a = Î fi = ( x − α i ) . Definition 2.6. (SRFR with Errors) Fix 0 < N , D , ε < f ≤ q . Aninstance of the SRFR with errors (SRFRwE) is a matrix ω ∈ K n × f whose columns are ω j = v ( α j )/ d ( α j ) + e j for some reduced v / d ∈ K ( x ) n × and some error matrix e . The reduced vector must satisfydeg ( v ) < N , deg ( d ) < D and d ( α i ) ,
0. The error matrix musthave its error support E : = { ≤ j ≤ f | e j , } which satisfies | E | ≤ ε .The solution of the SRFRwE instance ω is ( v , d ) . SRFRwE as Reed-Solomon code decoding.
We observe that if n = D = v / d is a polynomial. Then the SRFRwE is the problem ofrecovering a polynomial v given evaluations, some of which possi-bly erroneous. So in this case, SRFRwE is the problem of decodingan instance of a Reed-Solomon code .Its vector generalization, that is n > D =
1, coincides withthe decoding of an homogeneous Interleaved Reed-Solomon (IRS) code .Indeed, an IRS codeword can be seen as the evaluation of a vectorof polynomials v on α . Thus decoding IRS codes is the problem ofrecovering v from ω j = v ( α j ) + e j .Let us now detail how we can solve SRFRwE using SRFR. We usethe same technique of decoding RS and IRS codes [BW86, BKY03,PR17]. We introduce the Error Locator Polynomial Λ = Î j ∈ E ( x − α j ) . Its roots are the erroneous evaluations so deg ( Λ ) = | E | ≤ ε . Weconsider the Lagrangian polynomials u i ∈ K [ x ] such that u i ( α j ) = ω ij for any 1 ≤ i ≤ n . The classic approach is to remark that ( φ , ψ ) = ( Λ ( x ) v ( x ) , Λ ( x ) d ( x )) is a solution of φ = ψ u mod f Ö i = ( x − α i ) . (5)In order to reconstruct ( v , d ) it suffices to study the set of ( φ , ψ ) which verify Equation (5) and such that deg ( φ ) < N + ε and deg ( ψ ) < D + ε . In this way we reduce SRFRwE to SRFR (see Eq. 3). Hence,if f = ( N + ε ) + ( D + ε ) − = N + D + ε − cf. [BK14, KPSW17]).It is possible to reduce the number of evaluations w.r.t. the max-imal number of errors ε in the setting of IRS decoding ( D = Theorem 2.7 ([BKY03, BMS04, SSB09]).
Fix < N , ε < f ≤ q and E such that | E | ≤ ε . If f = N − + ε + ε / n , then for all ( v , ) and almost all error matrices e of support E , the SRFRwE admits aunique solution on the instance ω where ω j = v ( α j )/ d ( α j ) + e j . We prove a similar result in the rational function case,
Theorem 2.8 ([GLZ19, GLZ20]).
Fix < N , D , ε < f ≤ q and E such that | E | ≤ ε . If f = N + D − + ε + ε / n , then for all ( v , d ) and almost all error matrices e of support E , the SRFRwE admits aunique solution on the instance ω where ω j = v ( α j )/ d ( α j ) + e j . Since the problem of SRFRwE reduces to a simultaneous ratio-nal function reconstruction, the Equation (5) always admits a non-trivial solution whenever f = N + ε + ( D + ε − )/ n . Our idealresult would be to prove a uniqueness result also in this case. Ourexperiments suggest the following, Conjecture 2.9.
Fix < N , D , ε < f ≤ q and E such that | E | ≤ ε . If f = N + ε + ( D + ε − )/ n , then for almost all ( v , d ) andalmost all error matrices e of support E , the SRFRwE admits a uniquesolution on the instance ω where ω j = v ( α j )/ d ( α j ) + e j . Note that Conjecture 2.5 is for almost all fractions ( v , d ) whereasTheorems 2.7 and 2.8 are for all fractions. This difference is due toCounterexample 2.2, which states that we can not have uniquenessfor all ( v , d ) when f = N + ( D − )/ n . This latter number of evalu-ations matches the one of Conjecture 2.5 in the situation withouterrors ε =
0. Remark that this obstruction does not affect Theo-rems 2.7 and 2.8 because their number of evaluations f becomes N + D − ε = ω j , it re-mains to prove the existence of an instance of the form v ( α j )/ d ( α j ) + e j for any E such that | E | ≤ ε to prove the conjecture.The SRFRwE was first introduced by [BK14] in a special caseof its application, i.e. the Polynomial Linear System Solving withErrors, that we will introduce in the following paragraph. Polynomial linear system solving with errors.
We now supposethat we want to compute the unique solution of a PLS y ( x ) = v ( x )/ d ( x ) = A − b ∈ K [ x ] n × n in a scenario where some errorsoccur [BK14, KPSW17, GLZ19]. In detail, we fix f distinct evalua-tion points α = { α , . . . , α f } such that d ( α i ) ,
0. In our model, wesuppose that there is a black box which for any evaluation point α i , gives a solution of the evaluated systems of linear equations, i.e. y i = A ( α i ) − b ( α i ) . However, this black box could do some errorsin the computations. In particular, an evaluation α i is erroneous if y i , v ( α i )/ d ( α i ) and we denote by E : = { i | y i , v ( α i )/ d ( α i )} the set of erroneous positions. We refer to the problem of recon-structing the solution of a PLS in this model of errors as Polyno-mial Linear System Solving with Errors (shortly PLSwE). We ob-serve that if i ∈ E , then there exists a nonzero e i ∈ K n × f suchthat y i = v ( α i )/ d ( α i ) + e i . Hence, this problem is a special case leonora Guerrini, Romain Lebreton, Ilaria Zappatore of SRFRwE. Here we want to reconstruct a vector of rational func-tions which is a solution of a polynomial linear system. Therefore,all the results about uniqueness of the previous sections hold. Fur-thermore, in [KPSW17] authors introduced another bound whichguaranties the uniqueness based on the bounds on the degree ofthe polynomial matrix A and the vector b . In this section we will give some definitions and set out the nota-tion that we will use throughout this paper. We refer to [Nei16]for the definitions and lemmas of this section, and for historicalreferences. K [ x ] -module Let K be a field and K [ x ] the ring of polynomials over K . We startby defining the row degree of a vector, then of a matrix. Let p = ( p , . . . , p ν ) ∈ K [ x ] ν = K [ x ] × ν and s = ( s , . . . , s ν ) ∈ Z ν a shift. Definition 3.1 (Shifted row degree).
Let r i = deg ( p i ) + s i for 1 ≤ i ≤ ν . The s -row degree of p is rdeg s ( p ) = max ≤ i ≤ ν ( r i ) .We also denote p = ([ r ] s , . . . , [ r ν ] s ν ) a vector of polynomialswhere r i = deg ( p i ) + s i .We can extend this definition to polynomial matrices. In fact, let P ∈ K [ x ] ρ × ν be a polynomial matrix, with ρ ≤ ν . Let P j , ∗ be the j -th row of P for 1 ≤ j ≤ ρ . We can define the s -row degrees of thematrix P as rdeg s ( P ) : = ( r , . . . , r ρ ) where r j : = rdeg s ( P j , ∗ ) .Let N be a K [ x ] -submodule of K [ x ] ν = K [ x ] × ν . Since K [ x ] is a principal ideal domain, N is free of rank ρ : = rank (N) lessthan ν [DF03, Section 12.1, Theorem 4]. Hence, we can considera basis P ∈ K [ x ] ρ × ν , i.e. a full rank polynomial matrix, such that N = K [ x ] × ρ P = { λ P | λ ∈ K [ x ] × ρ } .Our goal is to define a notion of row degrees of N in order tostudy later the K -vector space N < r : = (cid:8) p ∈ N (cid:12)(cid:12) rdeg s ( p ) < r (cid:9) forsome r ∈ N . Different bases P of N have different s -row degreesso we need more definitions. We start with row reduced bases.Let t = ( t , . . . , t ν ) ∈ Z ν . We denote by X t a diagonal matrixwhose entries are x t , . . . , x t ν . Definition 3.2 (Shifted Leading Matrix).
The s -leading matrix of P is a matrix in K ρ × ν , whose entries are the coefficient of degreezero of X − rdeg s ( P ) PX s . Definition 3.3. (Row reduced basis) A basis P ∈ K [ x ] ρ × ν of N is s -row reduced (shortly s -reduced) if its leading matrix LM s ( P ) hasfull rank.This definition is equivalent to [Nei16, Definition 1.10], whichimplies that all s -reduced basis of N have the same row degree, upto permutation. We now focus on the following crucial property. Proposition 3.4. (Predictable degree property) P is s -reduced if and only if for all λ = ( λ , . . . , λ ρ ) ∈ K [ x ] × ρ ,rdeg s ( λ P ) = max ≤ i ≤ ρ ( deg ( λ i ) + rdeg s ( P i , ∗ )) = rdeg d ( λ ) where d = rdeg s ( P ) . The proof of this classic proposition can be found for instancein [Nei16, Theorem 1.11]. This latter proposition is useful because it implies that dim K N < r = Í { i | r i < r } ( r − r i ) where ( r , . . . , r ρ ) isthe s -row degree of any s -reduced basis of N .Since we will need to define the s -row degrees of N uniquely,not just up to permutation, we need to introduce ordered weakPopov form, which relies on the notion of pivot. Definition 3.5 (Pivot).
Let p ∈ K [ x ] × ν . The s -pivot index of p ismax { j | rdeg s ( p ) = deg ( p j ) + s j } . Moreover the corresponding p j is the s -pivot entry and deg ( p j ) is the s -pivot degree of p .We can naturally extend the notion of pivot to polynomial ma-trices. Definition 3.6. ((Ordered) weak Popov form) The basis P of N in s -weak Popov form if the s -pivot indices of its rows are pairwisedistinct. On the other hand, it is in s -ordered weak Popov form if thesequence of the s -pivot indices of its rows is strictly increasing.A basis in s -weak Popov form is s -reduced. Indeed, LM s ( P ) be-comes, up to row permutation, a lower triangular matrix with non-zero entries on the diagonal. Hence it is full-rank.Assume from now on that N is a submodule of K [ x ] ν of rank ν and that P is a basis of N in s -ordered weak Popov form. Then itspivot indices must be { , . . . , ν } .Weak Popov bases have a strong degree minimality property,stated in the following lemma. Lemma 3.7 ([Nei16, Lemma 1.17]).
Let s ∈ Z ν , P be a basis of N in s -weak Popov form with s -pivot degrees ( d , . . . , d ν ) . Let p ∈ N whose pivot index is ≤ i ≤ ν . Then the s -pivot degree of p is ≥ d i or equivalently rdeg s ( p ) ≥ rdeg s ( P i , ∗ ) . As it turns out, ordered weak Popov basis are reduced basis forwhich the s -row degree is unique. The following lemma is a con-sequence of Lemma 3.7. Lemma 3.8 ([Nei16, Lemma 1.25]).
Let s ∈ Z ν and assume N is a submodule of K [ x ] ν of rank ν . Let P and Q be two bases of N in s -ordered weak Popov form. Then P and Q have the same s -rowdegrees and s -pivot degrees. In this section, we will focus on the relation between pivots of weakPopov bases and leading terms w.r.t. a specific monomial order, asin Gröbner basis theory (see for instance [CLO98]).Let K [ x ] : = K [ x , . . . , x n ] be the ring of multivariate polynomi-als. Recall that a monomial in K [ x ] is a product of powers of theindeterminates x i : = x i · · · x i n n for some i : = ( i , . . . , i n ) ∈ N n .On the other hand, a monomial in K [ x ] n is x i ε j , where ε , . . . , ε n is the canonical basis of the K [ x ] -module K [ x ] n .A monomial order on K [ x ] n is a total order ≺ on the monomialsof K [ x ] n such that, for any monomials φ ε i , ψ ε j ∈ K [ x ] n and anymonomial τ , τ ∈ K [ x ] , φ ε i ≺ ψ ε j = ⇒ φ ε i ≺ τφ ε i ≺ τψ ε j . Given a monomial order ≺ on K [ x ] n and f ∈ K [ x ] n , the ≺ -initialterm in ≺ ( f ) of f is the term of f whose monomial is the greatestwith respect to the order ≺ . We remark that in the case of K [ x ] ,the only monomial order must be the natural degree order x a < x b ⇐⇒ a < b . n the Uniqueness of Simultaneous Rational Function Reconstruction Definition 3.9. (shifted-TOP order) Let ≺ be a monomial orderon K [ x ] . We consider the K [ x ] -module K [ x ] n with its canonicalbasis ε , . . . , ε n and let γ , . . . , γ n be monomials in K [ x ] . Then ≺ in-duces the following monomial order on K [ x ] n called s -TOP (TermOver Position): φ ε i ≺ s − T OP ψ ε j ⇐⇒ ( φγ i ≺ ψγ j ) or ( φγ i = ψγ j and i < j ) for any pairs of monomials φ ε i and ψ ε j of K [ x ] n .As for the univariate module K [ x ] n , the only monomial order ≺ on K [ x ] is the natural one. The shifting monomials are x s i , definedby the shift s = ( s , . . . , s n ) ∈ N n . Hence, the s -TOP order on K [ x ] n is x a ε i < s -TOP x b ε j ⇐⇒ ( a + s i , i ) ≺ lex ( b + s j , j ) (6)where ≺ lex is the lexicographic order on Z .We can now state the link between this monomial order andthe pivot’s definition: let p ∈ K [ x ] × n and in ≺ s -TOP ( p ) = αx d ε i be the ≺ s − T OP -initial term of p , then the s -pivot index, entry, anddegree are respectively i , p i and d . This will be useful later on, in e.g. Proposition 4.3.
Fix m ≥ n ≥
0, and M ∈ K [ x ] m × n . We consider a K [ x ] -submodule M of K [ x ] n . We define the K [ x ]− module homomorphism ˆ φ M : K [ x ] m −→ K [ x ] n /M p p M . Set A M , M : = ker ( ˆ φ M ) to get the injection φ M : K [ x ] m /A M , M ֒ → K [ x ] n /M . We call A M , M the relation module because p ∈ A M , M ⇔ φ M ( p ) = p M = M , i.e. p is a relation between rows of M .Let ε , . . . , ε m be the canonical basis of K [ x ] m , ε ′ , . . . , ε ′ n the canonical basis of K [ x ] n and e i ≡ ε i mod K [ x ] m / A M , M for 1 ≤ i ≤ m . Remark . We observe that by the
Invariant Factor Form of mod-ules over Principal Ideal Domains ( cf. [DF03, Theorem 4, Chapter12]), K : = K [ x ] n /M ≃ K [ x ] n / (cid:10) a i ( x ) ε ′ i (cid:11) ≤ i ≤ n for nonzeros a i ( x ) ∈ K [ x ] such that a n ( x )| a n − ( x )| . . . | a ( x ) . The polynomials a i ( x ) arethe invariants of the module M . We also denote f i : = deg ( a i ( x )) and we observe that f ≥ f ≥ . . . ≥ f n .From now on we will assume that M = (cid:10) a i ( x ) ε ′ i (cid:11) ≤ i ≤ n . Itmeans that any q ∈ K can be seen as ( q mod a , . . . , q n mod a n ) .Using the result of Lemma 3.8, we can define the row and pivotdegrees of the relation module A M , M . Definition 4.2 (Row and pivot degrees of the relation module).
Let s ∈ Z m be a shift and P be any basis of A M , M in ordered weakPopov form. The s -row degrees of the relation module A M , M are ρ : = rdeg s ( P ) = ( ρ , . . . , ρ m ) and the s -pivot degrees are δ : = ( δ , . . . , δ m ) where δ i = ρ i − s i .Throughout this paper we will also denote ρ M and δ M whenwe want to stress out the matrix dependency. In this section, we will see that the row degrees of the relation mod-ule can be deduced from the row rank profile of a matrix associated to ˆ φ M . We start by associating the pivot degree of p ∈ A M , M tolinear dependency relation. Proposition 4.3.
There exists p ∈ A M , M with s -pivot index i and s -pivot degree d if and only if x d e i ∈ B ≺ x d ε i M where B ≺ x d ε i M : = h x n e j | x n ε j ≺ s − T OP x d ε i i . Proof.
Fix i , d ∈ N and let p ∈ K [ x ] n with s -pivot index i and s -pivot degree d , so r : = rdeg s ( p ) = d + s i . Then p = ([≤ r ] s , . . . , [≤ r ] s i − , [ r ] s i , [ < r ] s i + , . . . , [ < r ] s m ) (see Definition 3.1) and we canwrite p = cx d ε i + p ′ where c ∈ K ∗ and p ′ = ([≤ r ] s , . . . , [≤ r ] s i − , [ < r ] s i , [ < r ] s i + , . . . , [ < r ] s m ) . So p ∈ A M , M has s -pivotindex i and degree d ⇔ x d ε i = − / c p ′ mod A M , M ⇔ x d e i ∈ (cid:28) x n e j (cid:12)(cid:12)(cid:12)(cid:12) n + s j ≤ d + s i , for 1 ≤ j ≤ i − n + s j < d + s i , for i ≤ j ≤ m (cid:29) = B ≺ x d ε i M . (cid:3) Theorem 4.4.
Let δ be the s -pivot degrees of the relation module A M , M . Then δ j = min { d | x d e j ∈ B ≺ x d ε j M } for any ≤ j ≤ m . Proof.
Fix 1 ≤ j ≤ m . During this proof we denote δ j : = min { d | x d e j ∈ B ≺ x d ε j M } . We want to prove that δ j = δ j . Recallthat by Proposition 4.3, x δ j e j ∈ B ≺ x δj ε j M . Hence, by the minimalityof δ j , δ j ≥ δ j . On the other hand, x δ j e j ∈ B ≺ x δj ε j M so by Propo-sition 4.3 there exists p ∈ A M , M of s -pivot index j and degree δ j .Finally, by Lemma 3.7 we can conclude that δ j ≥ δ j . (cid:3) We now define the ordered matrix Mo M as the matrix of ˆ φ M w.r.t. particular K -vector space bases: the rows of Mo M from topto bottom are the monomials of K [ x ] m sorted increasingly for the ≺ s − T OP order (see Eq. (6)). The columns of Mo M are written w.r.t. the basis { x i ε ′ j } ≤ j ≤ n ≤ i < f j of K [ x ] n /M . Therefore, Mo M has finiterank rank ( Mo M ) = rank ( ˆ φ M ) = rank ( φ M ) , infinite number ofrows and ( Í ni = f i ) = dim K ( K [ x ] n /M) columns. Monomial row rank profile.
Our goal is to relate the row rankprofile of Mo M to the row degree of the relation module. The clas-sic definition of row rank profile of a rank r polynomial matrix isthe lexicographically smallest sequence of r indices of linearly in-dependent rows ( cf. [DPS15] for instance). Since the rows of ourordered matrix Mo M correspond to monomials, we will transposethe previous definition to monomials instead of indices.Let Mon r be the sets of r monomials of K [ x ] m . We define thelexicographical ordering on Mon r by comparing lexicographicallythe sorted monomials for ≺ s − T OP . In detail, F < lex F ′ iff thereexists 1 ≤ t ≤ r s.t. x i l ε j l = x u l ε v l for l < t and x i t ε j t ≺ s − T OP x u t ε v t where F = { x i l ε j l } ≤ l ≤ r and F ′ = { x u l ε v l } ≤ l ≤ r andboth { x i l ε j l } and { x u l ε v l } are increasing for the ≺ s − T OP order.We will use this lexicographic order on monomials to define therow rank profile of Mo M . Let r = rank ( Mo M ) . Definition 4.5 (Row rank profile).
For any matrix M ∈ K [ x ] m × n ,we define the row rank profile of Mo M (shortly RRP M ) as the familyof monomials of K [ x ] m defined by RRP M : = min < lex P M where P M : = (cid:8) F ∈
Mon r (cid:12)(cid:12) { mM } m ∈F are linearly independent in K (cid:9) . leonora Guerrini, Romain Lebreton, Ilaria Zappatore We now introduce a particular family of monomials, that wewill frequently use: we will denote F d : = { x i ε j } i < d j ≤ j ≤ m for any d = ( d , . . . , d m ) ∈ N m .This family allows us to finally relate the row rank profile of Mo M to the row degree of the relation module. Proposition 4.6.
The row rank profile of the ordered matrix Mo M is given by the pivot degrees δ M of the relation module A M , M , i.e. RRP M = F δ M . Proof.
We fix the matrix M in order to simplify notations. Wedefine δ ′ j = min n δ | x δ ε j < RRP o and δ ′ = ( δ ′ , . . . , δ ′ m ) . By prop-erties of row rank profile, we have that x δ j e j ∈ B ≺ x δj ε j (otherwisewe could create a smaller family of linearly independent monomialwith x δ j e j ). Using Theorem 4.4, we deduce that δ ′ j ≥ δ j . There-fore F δ ⊂ F δ ′ ⊂ RRP . Since the families of monomials F δ and RRP have the same cardinality r = rank ( Mo ) , they are equal so F δ = RRP . (cid:3) We will now focus on integer tuples δ M which can be achieved. Forthis matter, in the light of Proposition 4.6, we need to understandwhich families F d of monomials can be linearly independent inthe ordered matrix, i.e. belong to P M (see Definition 4.5).Recall that K = K [ x ] n /M = K [ x ] n / (cid:10) a i ( x ) ε ′ i (cid:11) ≤ i ≤ n and f i = deg ( a i ( x )) are non-increasing as in Remark 4.1. Recall also fromDefinition 4.5 that P M is the set of families F of r monomials in K [ x ] m such that { mM } m ∈F are linearly independent in K [ x ] n /M . Theorem 4.7.
Let d ∈ N m be non-increasing. We can extend f ∈ N m by f n + = . . . = f m = . Then ∃ M ∈ K [ x ] m × n such that F d ∈ P M if and only if Í li = d i ≤ Í li = f i for all ≤ l ≤ m . The non-increasing property of d can be lifted: let d be non-increasing and d ′ be any permutation of d . Then ∃ M ∈ K [ x ] m × n such that F d ∈ P M if and only if ∃ M ′ ∈ K [ x ] m × n such that F d ′ ∈P M ′ . Indeed, permuting d amounts to permuting the componentsof p , i.e. permuting the rows of M . This does not affect the existenceproperty.The latter proposition is an adaptation of [Vil97, Proposition6.1] and its derivation [PS07, Theorem 3]. Even if the statements ofthese two papers are in a different but related context, their proofcan be applied almost straightforwardly. We will still provide themain steps of the proof, for the sake of clarity and also because wewill have to adapt the proof later in Theorem 5.2. Note also thatwe complete the ’if’ part of the proof because it was not detailedin earlier references. For this matter, we introduce the following Lemma 4.8.
Let N be a K [ x ] -submodule of K of rank l . Then thedimension of N as K -vector space is at most f + . . . + f l . Proof.
First, remark that if q ∈ N has its first non-zero elementat index p then a p ( x ) q =
0. Now since N has rank l , we can con-sider the matrix B whose rows are the l elements of a basis of N . Weoperate on the rows of B to obtain the Hermite normal form B ′ of B .The rows ( b ′ i ) ≤ i ≤ l of B ′ have first non-zero elements at distinctindices k , . . . , k l . Therefore a k j ( x ) b ′ j = { x i b ′ j } ≤ i < f kj ≤ j ≤ l is a generating set of N and so dim K N ≤ f k + . . . + f k l ≤ f + . . . + f l since ( f i ) are non increasing and ( k j ) pairwise distinct. (cid:3) Corollary 4.9.
Let r ≥ , d ∈ N l and v , . . . , v l ∈ K such that { x j v i } ≤ j < d i ≤ i ≤ l are linearly independent then Í li = d i ≤ Í li = f i . Proof.
We consider N the K [ x ] -module spanned by { v , . . . , v l } ,and we observe that d + . . . + d l ≤ dim N ≤ f + . . . + f l byLemma 4.8. (cid:3) Proof of Theorem 4.7.
We observe that if m > n , we can write K = K [ x ] n / (cid:10) a i ( x ) ε ′ i (cid:11) ≤ i ≤ n = K [ x ] m /h a i ( x ) ε i i ≤ i ≤ m where a j ( x ) = n + ≤ j ≤ m . Hence we can suppose w.l.o.g. that m = n . ⇒) By the hypotheses, there exists a matrix M ∈ K [ x ] m × n suchthat { x i ε j M } x i ε j ∈F d = { x i v j } < i < d j are linearly independent in K where v j : = ε j M . Hence, for all 1 ≤ l ≤ m , v , . . . , v l satisfythe conditions of the Corollary 4.9 and so Í li = d i ≤ Í li = f i . ⇐) Set u i = ε i for 1 ≤ i ≤ m so that { x i u j } i < f j ≤ j ≤ m are linearlyindependent in M . We now consider the matrix K : = [ K | . . . | K m ] where K j ∈ K [ x ] m × f j is in the Krylov form, that is K j = K ( u j , f j ) : = [ u j | x u j | . . . | x f j − u j ] by considering u j as a column vector. Notethat K is full column rank by construction. Our goal is to findvectors v , . . . , v m such that [ K ( v , d )| . . . | K ( v m , d m )] is full col-umn rank (see e K later).For this matter, we first need to consider the matrix K made ofcolumns of K so that it remains full column rank. It is defined as K : = [ K | . . . | K m ] where for 1 ≤ j ≤ m , K j ∈ K [ x ] m × d j aredefined iteratively by K j : = [ K ( u j , min ( f j , d j ))| K ( x s u j , t )| . . . | K ( x s k u j k , t k )] and K ( x s l u j l , t l ) derives from previously unused columns in K ,which we add from left to right, i.e. ( j l ) are increasing. Since Í ji = d i ≤ Í ji = f i , we will only pick from previous blocks, i.e. j k < j . Sincewe must have depleted a block K i l before going to another one, wecan observe that s l + t l = f l for l < k . The last block K i k is theonly one that may not be exhausted, i.e. s k + t k ≤ f k . Conversely, s l = d l for l > j l , except maybe the first block j where s ≥ d .We want to transform K j into a Krylov matrix e K j , working blockby block. First we extend [ K ( u j , min ( f j , d j ))| | . . . | ] to the right to K ( u j , d j ) . Then we extend all blocks [ | . . . | | K ( x s l u j l , t l )| | . . . | ] to the left and the right to K ( x s ′ l u j l , d l ) where s ′ l equals s l minus thenumber of columns of the left extension. In this way, the extensionmatches the original matrix on its non-zero columns. Now we candefine e K : = [ e K | . . . | e K m ] , where e K j : = K ( v j , d j ) with v j : = u j + Í kl = x s ′ l u j l .A crucial point of the proof is to show that s ′ k ≥
0. But since d i are-non increasing, j l are increasing and j k < j , we get s l ≥ d j l ≥ d j k ≥ d j . As the number of columns of the left extension is at most d j , we can conclude s ′ k ≥ T such that e K = KT . So we can conclude that e K ,which is in the desired block Krylov form, is full column rank as is K , which concludes the proof. (cid:3) n the Uniqueness of Simultaneous Rational Function Reconstruction Example 4.10.
We illustrate the construction of the proof of The-orem 4.7 with example. Let m = n = f = ( , , ) extended to f = d = ( , , , ) . Remark that Í li = d i ≤ Í li = f i for all1 ≤ l ≤ m . Then K = K ( u , d ) , K = [ K ( u , f )| K ( x d u , d − f )] picks its missing column from the first unused column of K , K = K ( u , d ) , and K = [ K ( u , f ) = œ| K ( x d + u , f − ( d + )| K ( x d u , f − d )] picks its 3 missing columns first from the 2unused of K , then from the remaining one of K . Then the con-struction extends K to e K = K ( v i , d i ) where v = u = [ , , ] , v = u + x d −( d − ) u = [ x , , ] , v = u = [ , , ] and v = x d + u + x d −( f −( d + )) u = [ x , , x ] . Finally the matrix M of thestatement of Theorem 4.7 has its j -th row M j , ∗ equal to v j . ^ We now have all the cards in our hand to state the principal con-straint on the pivot degree δ M of the relation module A M , M when M varies in the set of matrices K [ x ] m × n such that rank ( Mo M ) = rank ( φ M ) is fixed. We will denote by d r the pivot degree corre-sponding to the constraint. Theorem 4.11.
Recall that f = ( f , . . . , f m ) are the degrees ofthe invariants of M where f i = for n + ≤ i ≤ m , and let r = rank ( Mo M ) . Then F δ M ≥ lex F d r where F d r = min < lex ( F d ∈ Mon r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∀ ≤ l ≤ m , l Õ i = d i ≤ l Õ i = f i ) (7) Proof.
We know from Proposition 4.6 that
RRP M = F δ M so { x i ε j M } i < δ j , M ≤ j ≤ m are linearly independent and Í mi = δ i , M = r . UsingTheorem 4.7, we get that Í li = δ i , M ≤ Í li = f i for all 1 ≤ l ≤ m .This means that F δ M belongs to the set whose minimum is F d r ,which implies our result. (cid:3) We observe that r = rank ( Mo M ) must satisfy 0 ≤ r ≤ Σ : = Í mi = f i = dim K K [ x ] n /M and that r = Σ is reachable since m ≥ n .Note also that d r is well-defined in Theorem 4.11 as long as 0 ≤ r ≤ Σ : = Í mi = f i because it is related to the minimum of a non-empty set. We will now show that this pivot degree constraint d Σ is attain-able by δ M for matrices M such that rank ( Mo M ) = rank ( φ M ) = dim K K [ x ] n /M in which case φ M becomes a bijection. More specif-ically, we will show that this is the case for almost all matrices M ∈ K [ x ] m × n . Corollary 4.12.
For a generic matrix M ∈ K [ x ] m × n , the pivotdegrees δ M of the relation module A M , M satisfy δ M = d Σ where Σ = Í ni = f i . Proof.
Since Í li = d Σ , i ≤ Í li = f i for all 1 ≤ l ≤ m , we de-duce from Theorem 4.7 that there exists M ∈ K [ x ] m × n such that { mM } m ∈F d Σ are linearly independent. So the Σ -minor correspond-ing to those lines is non-zero for this matrix M . We now considerthis Σ -minor as a polynomial R in the coefficients of M . This poly-nomial is then nonzero since it admits a nonzero evaluation.Now for any matrix M = ( m i , j ) such that R ( m i , j ) ,
0, the vec-tors { mM } m ∈F d Σ must be linearly independent, so rank ( Mo M ) = Σ . We have RRP M ≤ lex F d Σ because F d Σ ∈ P M (see Definition 4.5). Theorem 4.11 gives the other inequality, so F d Σ = RRP M = F δ M and δ M = d Σ . (cid:3) In this section, we will see that our definitionof the generic pivot degree d Σ in Eq. (7) has a simplified expressionin a wide range of settings. Set the notation s = max ( s ) . We will seethat under some assumptions the expected row degree p Σ : = d Σ + s has a nice form. Define p and u be the quotient and remainder ofthe Euclidean division Í mi = ( f i + s i ) = p · m + u . The expected niceform of the row degrees will be p : = ( p + , . . . , p + | {z } u times , p , . . . , p | {z } m − u times ) . (8)This nice form will appear the following conditions on f and s : p ≥ s (9) ∀ ≤ l ≤ m − , l Õ i = p i ≤ l Õ i = ( f i + s i ) (10) Theorem 4.13.
Let p as in Equation (8) , ant let f be non-increasingsuch that Equations (9) and (10) hold. Then p Σ = p . This nice form of row degree was already observed in particu-lar cases in different but related settings. To the best of our knowl-edge, it can be found in [Vil97, Proposition 6.1] for row degrees ofminimal generating matrix polynomial but with no shift, in [PS07,Corollary 1] for dimensions of blocks in a shifted Hessenberg formbut the link to row degree is unclear and no shift is discussed(shifted Hessenberg is not related to our shift s ), and in [JV05, afterEq. (2)] for kernel basis were m = n with no shifts. Proof.
Denote again Σ = Í ni = f i . Let F be the first Σ mono-mials of K [ x ] m for the ≺ s − T OP ordering. Let p = ( p + , . . . , p + , p , . . . , p ) be the candidate row degrees as in the theorem state-ment and d = p − s be the corresponding pivot degrees. Note thatEquation (9) implies that p ≥ s so d ∈ N m .First we show that Equation (9) implies F = F d . For the firstpart, in order to prove F = F d , we need to show that d i = min { d ∈ N | x d ε i < F } . We already know that d i ∈ N . We will need to studythe row degrees of the first monomials to conclude. The monomialsof K [ x ] m of s -row degree r ordered increasingly for ≺ s − T OP are [ x r − s i ε i ] for increasing 1 ≤ i ≤ m such that s i ≤ r . There are m such monomials when r ≥ s . The monomials of s -row degree lessthan s are { x i ε j } i + s j < s and their number is Í mi = ( s − s i ) . From thiswe can deduce that the row degree of the n -th smallest monomialis (cid:4) ( n − − Í mi = ( s − s i ))/ m (cid:5) + s = (cid:4) ( n − + Í mi = s i )/ m (cid:5) providedthat n ≥ Í mi = ( s − s i ) +
1. We can now remark that the ( Σ + ) -thsmallest monomial has s -row degree p . More precisely, the ( Σ + ) -th smallest monomial is the ( u + ) -th monomial of row-degree r , so F is equal to all monomials of row degree less than p and the first u monomials of row degree p . This proves d i = min { d ∈ N | x d ε i < F } and F = F d .Second we deduce from Equation (10) that for all 1 ≤ l ≤ m , Í li = d i = Í li = ( p i − s i ) ≤ Í li = f i , so F d r ≤ lex F d by Theo-rem 4.11 and finally F d r = F d because F is the smallest set of Σ monomials. (cid:3) leonora Guerrini, Romain Lebreton, Ilaria Zappatore Example 4.14.
Here we provide 3 examples of generic row pivot d Σ and row degree p Σ : Corollary 4.12 applies only to the first sit-uation because the second and third situations are made so thatEq. (9) and respectively Eq. (10) are not satisfied. Let m = n = s = ( , , ) so that s = Í ( s − s i ) = f = ( , , ) , so Í ( f i + s i ) = ∗ m + p Σ = ( , , ) from Eq. (8) and d Σ = ( , , ) . In the second situation, f = ( , , ) and Eq. (9) is notsatisfied. We use Theorem 4.13 to get d Σ = ( , , ) from Eq. (7) and p Σ = ( , , ) . Finally in the third situation, f = ( , , ) and Eq. (10)is not satisfied. We use Theorem 4.13 to get d Σ = ( , , ) fromEq. (7) and p Σ = ( , , ) . Let F , F , F be the respective familiesof monomial of the three situations. We picture these families inthe following table, where Mon are the first monomials for ≺ s − T OP
Mon ε X ε X ε ε X ε X ε X ε X ε ε rdeg s F • • • • • • •F • • •F • • • • • • • Recall the SRFR, defined in Section 2.1. In particular, a , . . . , a n ∈ K [ x ] with degrees f i : = deg ( a i ) and u : = ( u , . . . , u n ) ∈ K [ x ] n such that deg ( u i ) < f i and 0 < N i ≤ f i for 1 ≤ i ≤ n , 0 < D ≤ min ≤ i ≤ n { f i } . We want to reconstruct ( v , d ) = ( v , . . . , v n , d ) ∈ K [ x ] ×( n + ) such that v i ≡ du i mod a i , deg ( v i ) < N i , deg ( d ) < D . We consider M = h a i ( x ) ε ′ i i and we denote by S u the set oftuples which verify Eq. (3). Lemma 5.1.
For the shift s = (− N , . . . , − N n , − D ) ∈ Z n + , wehave ( v , d ) ∈ S u ⇔ ( v , d ) ∈ A M , R u with rdeg s (( v , d )) < , where R u : = (cid:20) Id n − u (cid:21) ∈ K [ x ] ( n + )× n (11) Proof.
Observe that ( v , d ) ∈ S u if and only if it satisfies theequation v − d u ≡ ( v , d ) R u ≡ M , that is ( v , d ) ∈ A M , R u ,and if it satisfies the degree conditions equivalent to rdeg s (( v , d )) = max { deg ( v ) − N , . . . , deg ( v n ) − N n , deg ( d ) − D } < (cid:3) So in order to study the solutions of the SRFR we introduce the s -row degrees ρ u : = ρ R u and the s -pivot indices δ u : = δ R u of A R u , M (see Definition 4.2). As remarked just after the predictabledegree property (Proposition 3.4),dim K S u = dim K ( A R u , M ) < = − Õ ρ u , i < ρ u , i . (12)We can now show our main theorem about uniqueness in SRFRfor generic instances u . Theorem 5.2.
Assume Í ni = f i = Í ni = N i + D − . Then forgeneric u = ( u , . . . , u n ) ∈ K [ x ] × n , the solution space S u has di-mension as K -vector space. Proof.
By the previous considerations (see Eq. (12)) it is suffi-cient to prove that for generic u ∈ K [ x ] n + , ρ u = ( , . . . , , − ) .First, we need to show that the generic s -row degree p Σ is theexpected nice form p = ( , . . . , , − ) ( p = − u = n = m − Í ( f j + s j ) = − · m + ( m − ) , see Eq. (8)). It remains to checkthat we verify the hypotheses of Theorem 4.13. By Equation (9), s ≤ − = p . By Equation (10), Í li = p i ≤ ≤ Í li = ( f i + s i ) for all0 ≤ l ≤ m − f i + s i ≥ ≥ p i for all i .It remains to show that there exists a matrix of the form R u which satisfies the genericity condition of Corollary 4.12. Hence,the genericity condition is a non-zero polynomial when evaluatedon matrices R u and finally we have our result for generic u .In order to do so, we show that the construction of the proof ofthe Theorem 4.7 provides a matrix of the form R u in our case. Inour case ( d , . . . , d n + ) = ( N , . . . , N n , D − ) and m = n +
1, where f n + =
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