On the Complexity of the Generalized MinRank Problem

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On the Complexity of the Generalized MinRank Problem

Jean-Charles Faug`ere

^a

Mohab Safey El Din

^a

Pierre-Jean Spaenlehauer

^b,a,

∗

aUniversit´e Paris 6, INRIA Paris-Rocquencourt, PolSys Project, CNRS, UMR 7606 UFR Ing´enierie 919, LIP6.

Case 169. 4, Place Jussieu, F-75252 Paris, France.

bComputer Science Department, University of Western Ontario, London, ON, Canada.

Abstract

We study the complexity of solving thegeneralized MinRank problem, i.e. computing the set of points where the evaluation of a polynomial matrix has rank at mostr. A natural algebraic representation of this problem gives rise to adeterminantal ideal: the ideal generated by all minors of sizer+1 of the matrix.

We give new complexity bounds for solving this problem using Gr¨obner bases algorithms under genericity assumptions on the input matrix. In particular, these complexity bounds allow us to identify families of generalized MinRank problems for which the arithmetic complexity of the solving process is polynomial in the number of solutions. We also provide an algorithm to compute a rational parametrization of the variety of a 0-dimensional and radical system of bi-degree(D,1). We show that its complexity can be bounded by using the complexity bounds for the generalized MinRank problem.

Key words: MinRank, Gr¨obner basis, determinantal, bi-homogeneous, structured algebraic systems.

1. Introduction

We focus in this paper on the following problem:

Generalized MinRank Problem: given a fieldK, an×mmatrixM whose entries are polynomials of degreeDinK[x1, . . . ,x_k], andr<min(n,m)an integer, compute the set of points at which the evaluation ofM has rank at mostr.

This problem arises in many applications and this is what motivates our study. In cryptology, the security of several multivariate cryptosystems relies on the difficulty of solving the classical MinRank problem (i.e. when the entries of the matrix are linear (Bettale et al., 2012; Faug`ere

∗ Corresponding author.

Email addresses:[email protected](Jean-Charles Faug`ere),[email protected](Mohab Safey El Din),[email protected](Pierre-Jean Spaenlehauer).

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et al., 2008; Kipnis and Shamir, 1999)). In coding theory, rank-metric codes can be decoded by computing the set of points where a polynomial matrix has rank less than a given value (Faug`ere et al., 2008; Ourivski and Johansson, 2002). In non-linear computational geometry, many inci- dence problems from enumerative geometry can be expressed by constraints on the rank of a matrix whose entries are polynomials of degree frequently larger than 1 (see e.g. (Macdonald et al., 2001; Sottile, 2002, 2003)). Also, in real geometry and optimization (Bank et al., 2010;

Greuet et al., 2011; Safey El Din and Schost, 2003) the critical points of a map are defined by the rank defect of its Jacobian matrix (whose entries have degrees larger than 1 most of the time in applications). Moreover, this problem is also underlying other problems from symbolic computation (for instance solving multi-homogeneous systems, see e.g. Faug`ere et al. (2011)).

The ubiquity of this problem makes the development of algorithms solving it and complexity estimates of first importance. WhenKis finite, the generalized MinRank problem is known to be NP-complete (Buss et al., 1999); thus one can consider this problem as a hard problem.

To study the Generalized MinRank problem, we consider the algebraic system of all the(r+ 1)-minors of the input matrix. Indeed, these minors simultaneously vanish on the locus of rank defect and hence give rise to a section of adeterminantal ideal.

Several solving tools can be used to solve this algebraic system by taking profit of the underlying structure. For instance, thegeometric resolutionin Giusti et al. (2001) can use the fact that these systems can be evaluated efficiently. Also, recent works on homotopy methods (Verschelde, 1999) show that numerical algorithms can solve determinantal problems.

In this paper, we focus on Gröbner bases algorithms. A representation of the locus of rank defect is obtained by computing a lexicographical Gröbner basis by using the algorithms F₅ (Faugère, 2002) and FGLM (Faugère et al., 1993). Indeed, experiments suggest that these algorithms take profit of the determinantal structure. The aim of this work is to give an explanation of this behavior from the viewpoint of asymptotic complexity analysis.

Related works

An important related theoretical issue is to understand the algebraic structure of the ideal J_r⊂K[U](whereUis the set of variables{u_1,1, . . . ,u_n,m}) generated by the(r+1)-minors of the matrix:

U =







u_1,1 . . . u_1,m ... . .. ... u_n,1 . . . u_n,m





 .

The idealJ_rhas been extensively studied during last decades. In particular, explicit formulas for its degree and for its Hilbert series are known (see e.g. Fulton (1997, Example 14.4.14) and Conca and Herzog (1994)), as well as structural properties such as Cohen-Macaulayness and primality (Hochster and Eagon, 1970, 1971).

In cryptology, Kipnis and Shamir (1999) have proposed a multi-homogeneous algebraic mod- eling which can be seen as a generalization of the Lagrange multipliers and is designed as follows: a polynomialn×mmatrixM∈K[X]^n×m(whereXdenotes the set of variables{x₁, . . . ,x_k}) has rank at mostrif and only if the dimension of its right kernel is greater thanm−r−1. Con- sequently, by introducingr(m−r)fresh variablesy_1,1, . . . ,y_r,m−r, we can consider the system of bi-degree(D,1)inK[x1, . . . ,x_k,y_1,1, . . . ,y_r,m−r]defined by

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M·







1 0 . . . 0

0 1 . . . 0

... . .. . .. ...

0 0 . . . 1

y_1,1 y_1,2 . . . y_1,m−r ... ... . .. ... y_r,1 y_r,2 . . . y_r,m−r







=0.

If(x1, . . . ,x_k,y1,1, . . . ,y_r,m−r)is a solution of that system, then the evaluation of the matrixM at the point(x1, . . . ,x_k)has rank at mostr.

In (Faugère et al., 2010), the case of square linear matrices is studied by performing a complexity analysis of the Gröbner bases computations. In particular, this investigation showed that the overall complexity is polynomial in the size of the matrix when the rank defectn−ris con- stant. This theoretical analysis is supported by experimental results. The proofs were complete when the system has positive dimension, but depended on a variant of a conjecture by Fröberg in the 0-dimensional case.

Main results

We generalize in several ways the results from (Faug`ere et al., 2010) where only the case of square linear matrices was investigated: our contributions are the following.

• We deal with non-square matrices whose entries are polynomials of degreeDwith generic coefficients; this is achieved by using more general tools than those considered in (Faug`ere et al., 2010) (weighted Hilbert series). This generalization is important for applications in geometry and optimization for instance.

• Whenn= (p−r)(q−r), the solution set of the generalized MinRank problem has dimension 0. In that case, our proofs in this paper do not rely on Fr¨oberg’s conjecture; this has been achieved by modifying our proof techniques and using more sophisticated and structural properties of determinantal ideals. This is important for applications in cryptology (see e.g. the sets of parameters A, B and C in the MinRank authentication scheme (Courtois, 2001)).

Our results are complexity bounds for Gr¨obner bases algorithms when the input system is the set of(r+1)-minors of an×mmatrixM, whose entries are polynomials of degreeDwith generic coefficients.

By generic, we mean that there exists a non-identically null multivariate polynomialhsuch that the complexity results hold when this polynomial does not vanish on the coefficients of the polynomials in the matrix. Therefore, from a practical viewpoint, the complexity bounds can be used for applications where the base fieldKis large enough: in that case, the probability that the coefficients ofM do not belong to the zero set ofhis close to 1.

We start by studying the homogeneous generalized MinRank problem (i.e. when the entries of Mare homogeneous polynomials) and by proving an explicit formula for the Hilbert series of the idealIrgenerated by the(r+1)-minors of the matrixM. The general framework of the proofs is the following: we consider the idealJ_r⊂K[U]generated by the(r+1)-minors of a matrix U = (ui,j)whose entries are variables. Then we consider the idealJg_r=J_r+hg1, . . . ,g_nmi ⊂ K[U,X], where the polynomialsg_iare quasi-homogeneous forms that are the sum of a linear form

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inK[U]and of a homogeneous polynomial of degreeDinK[X]. If some conditions on theg_iare verified, by performing a linear combination of the generators there exists f_1,1, . . . ,f_n,m∈K[X] such that

Jg_r=J_r+hu1,1−f_1,1, . . . ,u_n,m−f_n,mi.

Then we use the fact that J_r+hu_1,1−f_1,1, . . . ,un,m−fn,mi

∩K[X] =Irto prove that properties of generic quasi-homogeneous sections ofJ_rtransfer toIrwhen the entries of the matrix M are generic. This allows us to use results known about the idealJ_r to study the algebraic structure ofIr.

We study separately three different cases:

• k>(n−r)(m−r). Under genericity assumptions on the input, the solutions of the generalized MinRank problem are an algebraic variety of positive dimension. Recall that the complexity results were only proven forD=1 andn=min Faug`ere et al. (2010). We generalize here for anyD∈N.

• k= (n−r)(m−r). This is the 0−dimensional case, where the problem has finitely-many solutions under genericity assumptions. Recall that the results in Faug`ere et al. (2010) were only stated forD=1 andn=m, and they depended on a variant of Fr¨oberg’s conjecture. In this paper, we give complete proofs forD∈Nwhich do not rely on any conjecture.

• k<(n−r)(m−r). In the over-determined case, we still need to assume a variant of Fr¨oberg’s conjecture to generalize the results in Faug`ere et al. (2010).

In particular, we prove that, fork≥(n−r)(m−r), the Hilbert series ofIris the power series expansion of the rational function

HS_I_r(t) =detA_r(t^D)(1−t^D)^{(n−r)(m−r)} t^D(^r2)(1−t)^k , whereAr(t) is the r×r matrix whose (i,j)-entry is∑k m−i

k

n−j k

t^k. Assuming w.l.o.g. that m≤n, we also prove that the degree ofIris equal to

DEG(Ir) =D^{(n−r)(m−r)}

m−r−1

∏

i=0

i!(n+i)!

(m−1−i)!(n−r+i)!.

These explicit formulas permit to derive complexity bounds on the complexity of the problem.

Indeed, one way to get a representation of the solutions of the problem in the 0-dimensional case is to compute alexicographicalGröbner basis of the ideal generated by the polynomials. This can be achieved by using first theF₅algorithm (Faugère, 2002) to compute a Gröbner basis for the so-calledgrevlexordering and then use the FGLM algorithm (Faugère et al., 1993) to convert it into alexicographicalGröbner basis. The complexities of these algorithms are governed by the degree of regularity and by the degree of the ideal.

Therefore the theoretical results on the structure ofIr yield bounds on the complexity of solving the generalized MinRank problem with Gr¨obner bases algorithms. More specifically, whenk= (n−r)(m−r)and under genericity assumptions on the input polynomial matrix, we prove that the arithmetic complexity for computing a lexicographical Gr¨obner basis of Ir is upper bounded by

O n

r+1 m

r+1

Dreg+k k

ω

+k(DEG(Ir))³

, where 2≤ω≤3 is a feasible exponent for the matrix multiplication, and

Dreg=Dr(m−r) + (D−1)k+1.

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This complexity bound permits to identify families of Generalized MinRank problems for which the number of arithmetic operations during the Gr¨obner basis computations is polynomial in the number of solutions.

In the over-determined case (i.e.k<(n−r)(m−r)), we obtain similar complexity results, by assuming a variant of Fr¨oberg’s conjecture which is supported by experiments.

Finally, we show that complexity bounds for solving systems of bi-degree(D,1)can be obtained from these results on the generalized MinRank problem. We give an algorithm whose arithmetic complexity is upper bounded by

O

n_x+ny

n_y+1

D(nx+n_y) +1 n_x

ω

+n_x

Dⁿ^x

n_x+n_y n_x

3! ,

for solving systems ofn_x+nyequations of bi-degree(D,1)inK[x1, . . . ,x_n_x,y₁, . . . ,y_n_y]which are radical and 0-dimensional.

Organization of the paper

Section 2 provides notations used throughout this paper and preliminary results. In Section 3, we show how properties of the idealJ_r generated by the(r+1)-minors ofU transfer to the idealIr. Then, the case when the homogeneous Generalized MinRank Problem has non-trivial solutions (under genericity assumptions) is studied in Section 4. Section 5 is devoted to the study of the over-determined MinRank Problem (i.e. whenk<(n−r)(m−r)). Then, the complexity analysis is performed in Section 6. Some consequences of this complexity analysis are drawn in Section 7. Experimental results are given in Section 7.4 and applications to the complexity of solving bi-homogeneous systems of bi-degree(D,1)are investigated in Section 8.

Acknowledgments

This work was supported in part by the HPAC grant and the GeoLMI grant (ANR 2011 BS03 011 06) of the French National Research Agency.

2. Notations and preliminaries

LetKbe a field andKbe its algebraic closure. In the sequel,n,m,randkandDare positive integers withr<m≤n. Ford∈N, Mon(d,k)denotes the set of monomials of degreed in the polynomial ringK[x1, . . . ,x_k]. Its cardinality is # Mon(d,k) = ^d−1+k_d

.

We denote byathe set of parameters{a^(i,j)_t : 1≤i≤n,1≤j≤m,t∈Mon(D,k)}. The set of variables{u_i,_j: 1≤i≤n,1≤j≤m}(resp.{x₁, . . . ,x_k}) is denoted byU(resp.X).

For 1≤i≤n,1≤j≤m, we denote by f_i,j∈K(a)[X]a generic form of degreeD f_i,j=

∑

t∈Mon(D,k)

a^(i,j)_t t.

LetIr⊂K(a)[X]be the ideal generated by the(r+1)-minors of then×mmatrix

M =







f_1,1 . . . f_1,m ... . .. ... f_n,1 . . . f_n,m





 ,

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andJ_r⊂K(a)[U,X]be the determinantal ideal generated by the(r+1)-minors of the matrix

U =







u_1,1 . . . u_1,m ... . .. ... u_n,1 . . . u_n,m





 .

We defineIfras the idealJ_r+hui,j−f_i,ji1≤i≤n,1≤j≤m⊂K(a)[U,X]. Notice thatIfr=Ir+ hu_i,_j−f_i,ji1≤i≤n,1≤j≤m⊂K(a)[U,X]. Therefore,Ir=Ifr∩K(a)[X].

By slight abuse of notation, ifIis a proper homogeneous ideal of a polynomial ringK[X], we callHilbert seriesofIand we noteHSI∈Z[[t]]the Hilbert series of its quotient algebraK[X]/I with the grading defined by deg(xi) =1 for alli:

HSI(t) =

∑

d≥0

dim_K(K[X]_d/I_d)t^d,

whereK[X]d denotes the vector space of homogeneous polynomials of degreed andI_d=I∩ K[X]_d.

We calldimensionofIthe Krull dimension of the quotient ringK[X]/I.

Quasi-homogeneous polynomials.

We need to balance the degrees of the entries of the matrixU with the degrees of the entries ofM. This can be achieved by putting aweighton the variablesu_i,j, giving rise toquasi- homogeneous polynomials. A polynomial f ∈K[U,X] is called quasi-homogeneous (of type (D,1)) if the following condition holds (see e.g. Greuel et al. (2007, Definition 2.11, page 120)):

f(λ^Du_1,1, . . . ,λ^Du_n,m,λx₁, . . . ,λx_k) =λ^df(u1,1, . . . ,u_n,m,x1, . . . ,x_k).

The integerdis called the weight degree of f and denoted by wdeg(f).

An idealI⊂K[U,X] is called quasi-homogeneous (of type(D,1)) if there exists a set of quasi-homogeneous generators. In this case, we denote byK[U,X]_dtheK-vector space of quasi- homogeneous polynomials of weight degreed, andI_ddenote the setK[U,X]d∩I.

Proposition 1. Let I⊂K[U,X]be an ideal. Then the following statements are equivalent:

(1) there exists a set of quasi-homogeneous generators of I;

(2) the sets I_dare subspaces ofK[U,X]d, and I=^L_d∈_NI_d. Proof. See e.g. Miller and Sturmfels (2005, Chapter 8).

2

IfIis a quasi-homogeneous ideal, then itsweighted Hilbert serieswHSI(t)∈Z[[t]]is defined as follows:

wHSI(t) =

∑

d∈N

dim(K[U,X]d/Id)t^d.

3. Transferring properties fromJ_rtoIr

In this section, we prove that generic structural properties (such as the dimension, the structure of the leading monomial ideal,. . . ) of the idealIfrare the same as properties of the idealJ_r

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where several generic forms have been added. Hence several classical properties of the determinantal idealJ_r transfer to the idealIfr. For instance, this technique permits to obtain explicit forms of the Hilbert series of the idealIfr.

In the following, we denote bybandcthe following sets of parameters:

b= {b^(`)_t |t∈Mon(D,k),1≤`≤nm};

c= {c^(`)_i,_j |1≤i≤n,1≤j≤m,1≤`≤nm}.

Also,g₁, . . . ,g_nm∈K(b,c)[U,X]are generic quasi-homogeneous forms of type(D,1)and of weight degreeD:

g_`=

∑

t∈Mon(D,k)

b_t^(`)t+

∑

1≤i≤n 1≤j≤m

c^(`)_i,_ju_i,_j.

We letJg_r denote the idealJ_r+hg1, . . . ,g_nmi ⊂K(b,c)[U,X]. Here and subsequently, for a= (ai,j)∈K^nm(^D−1+kD ), we denote byϕathe following evaluation morphism:

ϕa: K[a] −→ K f(a1,1, . . . ,a_n,m) 7−→ f(a1,1, . . . ,a_n,m) Also, for(b,c)∈K^nm

(^D−1+kD )^+nm, we denote byψ_b,cthe evaluation morphism:

ψ_b,c: K[b,c] −→ K f(b,c)7−→ f(b,c)

By abuse of notation, we letϕa(fIr)(resp.ψ_b,c(gJ_r)) denote the idealJ_r+hu_i,_j−ϕ_a(f_i,_j)i ⊂ K[U,X](resp.J_r+hψb,c(g1), . . . ,ψb,c(gnm)i ⊂K[U,X]).

We callpropertya map from the set of ideals ofK[U,X]to{true,false}:

P: Ideals(K[U,X])→ {true,false}. Definition 2. LetPbe a property. We say thatPis

• Ifr-generic if there exists a non-empty Zariski open subsetO⊂K^nm(^D−1+kD )such that a∈O⇒P

ϕa

Ifr

=true;

• Jg_r-generic if there exists a non-empty Zariski open subsetO⊂K^nm

(^D−1+kD )^+nmsuch that (b,c)∈O⇒P

ψ_b,c

Jg_r

=true.

The following lemma is the main result of this section:

Lemma 3. A propertyPisIfr-generic if and only if it isJg_r-generic.

Proof. To obtain a representation ofϕ_a

Jg_r

for a genericaas a specialization ofIfr (and conversely), it is sufficient to perform a linear combination of the generators. The point of this proof is to show that genericity is preserved during this linear transform.

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In the sequel we denote byA,B andC the following matrices (of respective sizes nm×

D−1+k D

,nm× ^D−1+k_D

andnm×nm):

A=





 a⁽¹⁾

x^D₁ a⁽¹⁾

x^D−1₁ x₂ . . . a⁽¹⁾

x^D_k

... ... ... ... a^(nm)

x^D₁ a^(nm)

x^D−1₁ x2 . . . a^(nm)

x^D_k







B =





 b⁽¹⁾

x^D₁ b⁽¹⁾

x^D−1₁ x₂ . . . b⁽¹⁾

x^D_k

... ... ... ... b^(nm)

x^D₁ b^(nm)

x^D−1₁ x₂ . . . b^(nm)

x^D_k







C =







c⁽¹⁾_1,1 . . . c⁽¹⁾_n,m ... ... ... c^(nm)_1,1 . . . c^(nm)_n,m





 .

Therefore, we have







u_1,1−f_1,1 ... u_n,m−f_n,m







= Idnm·





 u_1,1

... u_n,m







−A·





 x^D₁ x₁^D−1x₂

... x^D_k











 g₁

... gnm







= C·





 u_1,1

... u_n,m





 +B·





 x^D₁ x^D−1₁ x2

... x^D_k







In this proof, fora∈K^nm(^D−1+kD )(resp.b∈K^nm(^D−1+kD ),c∈Kⁿ

2m²), the notationA(resp.B,C) stands for the evaluation of the matrixA(resp.B,C) ata(resp.b,c). Also, we implicitly identify Awitha(resp.Bwithb,Cwithc,Awitha,Bwithb,Cwithc).

• LetPbe aIfr-generic property. Thus there exists a non-zero polynomialh₁(A)∈K[a]such that ifh1(A)6=0 thenP

ϕa(fIr)

=true.

Let adj(C) denote the adjugate of C (i.e. adj(C) =det(C)·C⁻¹ in K(c)). Consider the polynomial he₁ defined by he₁(B,C) =h₁(−adj(C)·B)∈K[b,c]. The polynomial inequality det(C)he₁(B,C)6=0 defines a non-empty Zariski open subsetO⊂K^nm

(^D−1+kD )^+nm.Let (B,C)∈Obe an element in this set, thenCis invertible since det(C)6=0. LetAebe the matrix Ae=−adj(C)·B. Therefore the generators of the idealϕ

ea

Ifr

are an invertible linear combination of the generators ofψb,c

Jg_r

. Consequently,ϕ

ea

Ifr

=ψb,c

Jg_r

. Moreover,

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h₁(A) =e he₁(B,C)6=0 implies that the polynomialhe₁is not identically 0. Therefore,

∀(b,c)∈O,P ψ_b,c

Jg_r

=P ϕea

Ifr

=true,

and hencePis aJg_r-generic property.

• Conversely, consider aJg_r-generic property P. Thus, there exists a non-zero polynomial h₂(B,C)∈K[b,c] such that ifh₂(b,c)6=0 thenP

ψb,c(gJ_r)

=true. SinceP isJg_r- generic, there exists (b,c) such that h₂(b,c)det(c)6=0. Lethe₂ be the polynomialhe₂(b) = h2(−C·B,C).

Since det(C)6=0, the matrixCis invertible andhe2(−C⁻¹·B) =h2(B,C)6=0 and hence the polynomialhe₂is not identically 0. Moreover, ifa∈K^nm(^D−1+kD )is such thathe₂(A)6=0, then h2(−C·A,C)6=0 and thusP

ψ_−C·A,C(gJ_r)

=true. Finally, ψ_−C·A,C(gJ_r) =ϕA(fIr) since the generators ofψ_−C·A,C(Jg_r)are an invertible linear combination of that ofϕ_a(Ifr) (the linear transformation being given by the invertible matrixC) and hence they generate the same ideal. Therefore, the propertyPisIfr-generic.

2

In the sequel,≺is an admissible monomial ordering (see e.g Cox et al. (1997, Chapter 2,

§2, Definition 1)) onK[U,X], and for any polynomial f ∈K[U,X],LM(f)denotes its leading monomial with respect to≺. IfI is an ideal ofK[U,X],K(a)[U,X], orK(b,c)[U,X], we let LM(I)denote the ideal generated by the leading monomials of the polynomials.

By slight abuse of notation, ifI1andI2 are ideals ofK[U,X],K(a)[U,X], or K(b,c)[U,X] (I₁ andI₂ are not necessarily ideals of the same ring), we writeLM(I1) =LM(I2)if the sets {LM(f)| f∈I₁}and{LM(f)| f ∈I₂}are equal.

Lemma 4. LetP_I_f

r andP_J_g

r

be the properties defined by P_I_f_r(I) =

(trueif LM(I) =LM Ifr

; falseotherwise.

P_J_g

r(I) =

(trueif LM(I) =LM Jg_r

; falseotherwise.

ThenP_I_f_r (resp.P_J_g

r) is aIfr-generic (resp.Jg_r-generic) property.

Proof. We prove here thatP_I_f

r isIfr-generic (the proof forP_J_g

r

is similar).

The outline of this proof is the following: during the computation of a Gröbner basisGof Ifr inK(a)[U,X] (for instance with Buchberger’s algorithm), a finite number of polynomials are constructed. Let ϕabe a specialization. If the images byϕa of the leading coefficients of all non-zero polynomials arising during the computation do not vanish, thenϕa(G)⊂ϕa(fIr)is a Gröbner basis of the ideal it generates. It remains to prove thatϕ_a(G)is a Gröbner basis of ϕa(fIr). This is achieved by showing that generically, the normal form (with respect toϕa(G)) of the generators ofϕa(Ifr)is equal to zero.

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For polynomialsf₁,f₂, we letLC(f₁)(resp.LC(f₂)) denote the leading coefficient off₁(resp.

f₂) andSpol(f₁,f₂) =^LCM_LC⁽^LM₍_f⁽^f¹^),^LM⁽^f²⁾⁾

1)LM(f1) f₁−^LCM_LC⁽^LM₍_f^(f¹^),^LM⁽^f²⁾⁾

2)LM(f2) f₂denote theS-polynomialof f₁ andf₂.

We need to prove that there exists a non-empty Zariski open subsetO1⊂K^nm(^D−1+kD )such that a∈O₁⇒LM(ϕa(fIr)) =LM(fIr).

To do so, consider a Gr¨obner basisG⊂K(a)[U,X]ofIfr such that each polynomialgcan be written as a combinationg=∑h_`f_`, where the f_`’s range over the set of minors of sizer+1 ofU and the polynomialsu_i,j−f_i,_j, andh_`∈K[a][U,X]. Buchberger’s criterion states that S- polynomials of polynomials in a Gr¨obner basis reduce to zero (Cox et al., 1997, Chapter 2,§6, Theorem 6). Thus each S-polynomial ofg_i,g_j∈Gcan be rewritten as an algebraic combination

Spol(gi,g_j) =

∑

`

h⁰_`g_`,

where the polynomialsh⁰_`belongs toK(a)[U,X]and such that{g₁, . . . ,g_t_i,j} ⊂Gand for each 1≤s≤t_i,j,LM(gs)dividesLM(Spol(g,g⁰)−∑^s−1_`=1h⁰_`g_`). Next, consider:

• the productQ1(a) =∏g∈GLC(g)of the leading coefficients of the polynomials in the Gr¨obner basis;

• for all(gi,g_j)∈G²such thatSpol(gi,g_j)6=0, the productQ₂(a)of the numerators and denominators of the leading coefficients arising during the reduction ofSpol(gi,g_j).

These coefficients belongs toK[a]. Denote byQ(a) =Q1(a)Q2(a)∈K[a]their product. The inequalityQ(a)6=0 defines a non-empty Zariski open subsetO₁⊂K^nm(^D−1+kD ). Ifa∈O₁, then

ϕa(Spol(g,g⁰)) =

t

`=1

∑

ϕa(h⁰_`)ϕa(g_`),

and for each 1≤i≤t, LM(ϕa(gi)) divides LM(ϕa(Spol(g,g⁰))−∑ⁱ⁻¹_`=1ϕa(h⁰_`)ϕa(g`)). Thus ϕa(G)is a Gr¨obner basis of the ideal it spans. Moreover,hϕ_a(G)i ⊂ϕa(fIr).

We prove now that there exists a non-empty Zariski open set where the other inclusion ϕ_a(Ifr)⊂ hϕ_a(G)iholds. LetNFG(·)be the normal form associated to this Gr¨obner basis (as defined as theremainder of the division by Gin Cox et al. (1997, Chapter 2,§6, Proposition 1)). For each generator f ofIfr(i.e. either a maximal minor of the matrixU, or a polynomialu_i,_j−f_i,_j), we have thatNFG(f) =0. During the computation ofNFG(f)by using the division Algorithm in Cox et al. (1997, Chapter 2,§3), a finite set of polynomials (inK(a)[U,X]) is constructed.

LetQ₃∈K[a]denote the product of the numerators and denominators of all their nonzero coefficients. Consequently, ifQ⁽₃^f⁾(a)6=0, thenNF_ϕ_a_(G)(ϕa(f)) =0 and henceϕa(f)∈ hϕ_a(G)i.

Repeating this operation for all the generators ofIfr yields a finite set of non-identically null polynomials Q⁽₃^f⁾ ∈K[a]. Let Q₄∈K[a] denote their product. Therefore, ifQ₄(a)6=0, then ϕa(fIr)⊂ hϕ_a(G)i.

Finally, consider the non-empty Zariski open subsetO⊂K^nm(^D+k−1D )defined by the inequality Q₁·Q₂·Q₄6=0. For alla∈O, we haveϕa(Ifr) =hϕ_a(G)i.

2

Corollary 5. The leading monomials ofIfrare the same as that ofJg_r: LM

Ifr

=LM Jg_r

.

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Proof. By Lemmas 3 and 4, the propertyP_I_f

r (resp. P_J_g

r

) isIfr-generic and Jg_r-generic.

Since P_J_g

r

(resp.P_I_f

r) is Jg_r-generic, there exists a non-empty Zariski open subset O1⊂ K^nm

(^D−1+kD )^+nm(resp.O₂⊂K^nm

(^D−1+kD )^+nm) such that, for(b,c)∈O₁(resp.O₂),LM

ψ_(b,c)(gJ_r)

= LM

Jg_r

(resp.LM

ψ_(b,c)(Jg_r)

=LM Ifr

).

Notice thatO₁∩O₂is not empty, since for the Zariski topology, the intersection of finitely- many non-empty open subsets is non-empty. Let(b,c)be an element ofO₁∩O₂. Then

LM Ifr

=LM

ψ_(b,c)(gJ_r)

=LM Jg_r

. 2

Corollary 6. The weighted Hilbert series ofIfris the same as that ofJg_r.

Proof. It is well-known that, for any positively graded idealI and for any monomial ordering,wHSI(t) =wHSLM(I)(t)(see e.g. the proof of Cox et al. (1997, Chapter 9,§3, Proposition 9) which is also valid for quasi-homogeneous ideals). By Corollary 5,LM

Ifr

=LM Jg_r

, which implies that

wHS_LM

Ifr

(t) =wHS_LM

Jg_r(t), and hencewHS

Ifr(t) =wHS

Jg_r(t). 2 4. The casek≥(n−r)(m−r)

As we will see in the sequel, the Krull dimension of the ringK(a)[X]/Iris equal to max(k− (n−r)(m−r),0). This section is devoted to the study of the casek≥(n−r)(m−r).

We show here that the algebraic structure of the idealIris closely related to that of a generic section of a determinantal variety.

We recall that the polynomialsg_`are defined by g_`=

∑

t∈Mon(D,k)

b_t^(`)t+

∑

1≤i≤n 1≤j≤m

c^(`)_i,_ju_i,_j.

Lemma 7. Let1≤`≤nm be an integer. If g_`divides zero inK(b,c)[U,X]/ J_r+hg1, . . . ,g_`−1i , then there exists a prime ideal P associated toJ_r+hg1, . . . ,g_`−1isuch thatdim(P) =0.

Proof. Ifg_`divides zero inK(b,c)[U,X]/ J_r+hg1, . . . ,g_`−1i

, then there exists a prime ideal Passociated toJ_r+hg1, . . . ,g_`−1isuch thatg_`∈P. For`≤nm, letb^(≤`)andc^(≤`)denote the sets of parameters

b^(≤`) = {b^(s)_t |t∈Mon(D,k),1≤s≤`}

c^(≤`) = {c^(s)_i,_j |1≤i≤n,1≤j≤m,1≤s≤`}.

Since J_r+hg1, . . . ,g_`−1i

is an ideal ofK(b^(≤`−1),c^(≤`−1))[U,X], andPis an associated prime, there exists a Gr¨obner basisGP of P(for any monomial ordering≺) which is a finite subset ofK(b^(≤`−1),c^(≤`−1))[U,X].

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LetNFP(·)denote the normal form associated to this Gr¨obner basis (as defined as theremainder of the division by G_Pin Cox et al. (1997, Chapter 2,§6, Proposition 1)).

Sinceg_`∈P, we haveNFP(g`) =0. By linearity ofNFP(·), we obtain

t∈Mon(D,k)

∑

b^(`)_t NFP(t) +

∑

1≤i≤n 1≤j≤m

c^(`)_i,_jNFP(ui,j) =0.

Since G_p⊂K(b^(≤`−1),c^(≤`−1))[U,X], we can deduce that for any monomial t, NFP(t)∈ K(b^(≤`−1),c^(≤`−1))[U,X]. Therefore, by algebraic independence of the parameters, the following properties hold: for allt∈Mon(D,k),NFP(t) =0, and for alli,j,NFP(ui,j) =0. Consequently, all monomials of weight degreeDinK(b,c)[U,X]are inP, and hencePhas dimension 0. 2

Lemma 8. For all`∈ {2, . . . ,nm}, the polynomial g_`does not divide zero inK(b,c)[U,X]/(J_r+hg1, . . . ,g_`−1i) anddim(J_r+hg1, . . . ,g_`i) =k+ (n+m−r)r−`.

Proof. We prove the Lemma by induction on`. According to Hochster and Eagon (1970, Corol- lary 2 of Theorem 1), the ringK(b,c)[U,X]/J_r is Cohen-Macaulay and purely equidimensional. First, notice that the dimension is equal tok+ (n+m−r)rfor`=0 since the dimension of the idealJ_r⊂K[U]is(n+m−r)r(see e.g. Conca and Herzog (1994) and references therein). Now, suppose that the dimension of the ideal J_r+hg1, . . . ,g_`−1i ⊂K(b,c)[U,X] is k+ (n+m−r)r−`+1. Since the ringK(b,c)[U,X]/J_ris Cohen-Macaulay andhg₁, . . . ,g_`−1i has co-dimension`−1 inK(b,c)[U,X], the Macaulay unmixedness Theorem (Eisenbud, 1995, Corollary 18.14) implies that hg₁, . . . ,g_`−1i has no embedded component and is equidimensional inK(b,c)[U,X]/J_r. HenceJ_r+hg1, . . . ,g_`−1ias an ideal inK(b,c)[U,X]has no embedded component and is equidimensional. By contradiction, suppose thatg_`divides zero in

K(b,c)[U,X]/(J_r+hg1, . . . ,g_`−1i). By Lemma 7, there exists a primePassociated toJ_r+hg1, . . . ,g_`−1i such that dim(P) =0, which contradicts the fact thatJ_r+hg1, . . . ,g_`−1iis purely equidimen-

sional of dimensionk+ (n+m−r)r−`+1>0. 2

Lemma 9. The Hilbert series of theIr⊂K(a)[X]equals the weighted Hilbert series ofIfr⊂ K(a)[X,U].

Proof. Let≺_lex denote a lexicographical ordering onK(a)[X,U]such that x_k ≺_lexu_i,_j for all k,i,j. By Cox et al. (1997, Section 9.3, Proposition 9),HS_I_r(t) =HSLM≺lex(Ir)(t)andwHS

Ifr(t) = wHS_LM

≺lex(fIr)(t). SinceLM≺_lex(ui,j−f_i,j) =u_i,_j, we deduce that all monomials which are mul- tiples of a variableu_i,_j are inLM≺_lex(Ifr). Therefore, the remaining monomials inLM≺_lex(Ifr) are inK(a)[X]:

LM≺_lex(Ifr) = D

{u_i,j} ∪LM≺_lex(Ifr∩K(a)[X])E

=

{u_i,j} ∪LM≺_lex(Ir) . Therefore, ^K(a)[U,X]

LM≺lex(fIr) is isomorphic (as a gradedK(a)-algebra) to _LM^K(a)[X]_≺

lex(Ir). Thus HSLM≺lex(Ir)(t) =wHS_LM

≺lex(fIr)(t), and hence

HS_I_r(t) =wHS

Ifr(t).

2

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In the sequel,A_r(t)denotes ther×rmatrix whose(i,j)-entry is∑k m−i

k

n−j k

t^k. The following theorem is the main result of this section:

Theorem 10. The dimension of the idealIris k−(n−r)(m−r)and its Hilbert series is HS_I_r(t) =det A_r(t^D)

(1−t^D)^{(n−r)(m−r)} t^D(^r2)(1−t)^k .

Proof. According to Conca and Herzog (1994, Corollary 1) (and references therein), the ideal J_r seen as an ideal ofK[U]has dimension(m+n−r)rand its Hilbert series (for the standard gradation: deg(ui,j) =1) is the power series expansion of

HS_J

r⊂K[U](t) = detA_r(t) t(2^r)(1−t)^(n+m−r)r.

By putting a weightDon each variableu_i,_j(i.e. deg(ui,j) =D), the weighted Hilbert series of J_r⊂K[U]is

wHS_J

r⊂K[U](t) = detA_r(t^D) t^D(^r2)(1−t^D)^(n+m−r)r.

By consideringJ_ras an ideal ofK(b,c)[U,X], the dimension becomesk+ (m+n−r)rand its weighted Hilbert series is

wHS_J

r⊂K(b,c)[U,X](t) = detA_r(t^D)

t^D(2^r)(1−t)^k(1−t^D)^(n+m−r)r.

According to Lemma 8, for each`≤nm, the polynomialg_`does not divide zero in the ring K(b,c)[U,X]/(J_r+hg1, . . . ,g_`−1i). This implies the following relations:

dim J_r+hg1, . . . ,g_`i

= dim J_r+hg1, . . . ,g_`−1i

−1 wHS_J

r+hg1,...,g_`i(t) = (1−t^D)wHS_J

r+hg1,...,g_`−1i(t).

Therefore the dimension of Jg_r is k−nm+ (n+m−r)r and its quasi-homogeneous Hilbert series is

wHS

Jg_r(t) = det A_r(t^D)

t^D(^r2)(1−t)^k(1−t^D)(n+m−r)r−nm=det A_r(t^D)

(1−t^D)^{(n−r)(m−r)} t^D(2^r)(1−t)^k . By Corollary 6, the idealIfr has the same weighted Hilbert series. Finally, by Lemma 9, the Hilbert series ofIr=Ifr∩K(a)[X]is the same as that ofIfr. 2

Corollary 11. The degree of the idealIris:

DEG(Ir) = D^{(n−r)(m−r)}

m−r−1

∏

i=0

i!(n+i)!

(m−1−i)!(n−r+i)!

=D^{(n−r)(m−r)}

m−r−1

∏

i=0

n+m−r−1 r+i

n+m−r−1 i

.

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Proof. From Fulton (1997, Example 14.4.14), the degree of the idealJ_ris

m−r−1

∏

i=0

i!(n+i)!

(m−1−i)!(n−r+i)!.

Since the degree is equal to the numerator of the Hilbert series ofJ_revaluated att=1, detA_r(1) =

m−r−1

∏

i=0

i!(n+i)!

(m−1−i)!(n−r+i)!. By Theorem 10, the Hilbert series ofIris

HS_I_r(t) = det A_r(t^D)

(1−t^D)^{(n−r)(m−r)} t^D(2^r)(1−t)^k

= det A_r(t^D)

(1+t+· · ·+t^D−1)^{(n−r)(m−r)} t^D(2^r)(1−t)k−(n−r)(m−r) . Thus, the evaluation of the numerator int=1 yields

DEG(Ir) =D^{(n−r)(m−r)}

m−r−1

∏

i=0

i!(n+i)!

(m−1−i)!(n−r+i)!. To prove the second equality, notice that

m−r−1

∏

i=0

n+m−r−1 r+i

n+m−r−1 i

=

m−r−1

∏

i=0

i!(n+m−r−i−1)!

(r+i)!(n+m−2r−i−1)!. By substitutingibym−r−1−i, we obtain that

m−r−1

∏

i=0

(n+m−r−i−1)! =

m−r−1

∏

i=0

(n+i)!

m−r−1

∏

i=0

(r+i)! =

m−r−1

∏

i=0

(m−i−1)!

m−r−1

∏

i=0

(n+m−2r−i−1)! =

m−r−1

∏

i=0

(n−r+i)!.

Consequently,

m−r−1

∏

i=0

i!(n+i)!

(m−1−i)!(n−r+i)!=

m−r−1

∏

i=0

n+m−r−1 r+i

n+m−r−1 i

. 2

5. The over-determined case

To study the over-determined case (k<(n−r)(m−r)), we need to assume a variant of Fr¨oberg’s conjecture (Fr¨oberg, 1985):

Conjecture 12. LetJ_`,idenote the vector space of quasi-homogeneous polynomials of weight degree i inJ_r+hg1, . . . ,g_`i. Then the linear map

K(b,c)[U,X]i/J_`,i −→ K(b,c)[U,X]i+D/J_`,i+D

f 7−→ f g_`+1

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has maximal rank, i.e. it is either injective or onto.

Remark 13. Ifk+ (n+m−r)r−` >0, then Conjecture is proved by Lemma 8: g_`+1does not divide zero inK(b,c)[U,X]/ J_r+hg1, . . . ,g_`i

and hence the linear map is injective for all i∈N.

Notation.Given a power seriesS(t)∈Z[[t]], we let[S(t)]₊denote the power series obtained by truncatedS(t)at its first non positive coefficient.

Lemma 14. If Conjecture 13 is true, then the Hilbert series ofJ_r+hg1, . . . ,g_`+1iis wHS_J

r+hg1,...,g_`+1i(t) =h

(1−t^D)wHS_J

r+hg1,...,g_`i(t)i

+.

Proof. In this proof, for simplicity of notation, we letRdenote the ringK(b,c)[U,X]. IfS(t) =

∑i∈Ns_itⁱ∈Z[[t]]is a power series,[S(t)]_≥0denotes the series [S(t)]_≥0=

∑

i∈N

max(si,0)tⁱ.

Letann(g`+1)be the ideal{f∈R: f g_`+1∈J_r+hg1, . . . ,g_`i}. Fori∈N, consider the following exact sequence:

0→ann(g`+1)i→R_i/J_`,i−−−→^×g^`+1 R_i+D/J_`,i+D→

→R_i+D/J_`+1,i+D→0.

By Conjecture 13, we obtain

dim(ann(g`+1)i) =max(0,dim(Ri/J_`,i)−dim(Ri+D/J_`,i+D)).

The alternate sum of the dimensions of the vector spaces occurring in an exact sequence is zero;

it follows that

dim(Ri+D/J_`+1,i+D) = dim(Ri+D/J_`,i+D)−dim(Ri/J_`,i)+

max(0,dim(Ri/J_`,i)−dim(Ri+D/J_`,i+D))

= max(0,dim(Ri+D/J_`,i+D)−dim(Ri/J_`,i)).

Multiplying this identity byt^i+Dyields t^i+D

wHS_J

r+hg1,...,g_`+1i(t) = dim

R_i+D/J_`+1,i+D)

= max

0,dim(Ri+D/J_`,i+D)−dim(Ri/J_`,i)

= max

0,[t^i+D](1−t^D)wHS_J_r_+hg₁_,...,g_`i(t)

= [t^i+D]h

(1−t^D)wHS_J

r+hg₁,...,g_`i(t)i

≥0.

Since any monomial inK(a)[X,U]of weight degree greater thatDis a multiple of a monomial of weight degreeD, we deduce that if there existsi₀≥Dsuch that

tⁱ⁰ wHS_J

r+hg₁,...,g_`+1i(t) =0, then for alli>i₀,

tⁱ wHS_J

r+hg₁,...,g_`+1i(t) =0. Therefore t^i+D

wHS_J

r+hg₁,...,g_`+1i(t) = [t^i+D]h

(1−t^D)wHS_J

r+hg₁,...,g_`i(t)i

+,

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Finally, by summing overi, we get wHS_J

r+hg1,...,g_`+1i(t) =h

(1−t^D)HS_J

r+hg1,...,g_`i(t)i

+. 2

Theorem 15. If Conjecture 13 is true, then the Hilbert series ofIris HS_I_r(t) =

"

(1−t^D)^{(n−r)(m−r)}det A_r(t^D) t^D(2^r)(1−t)^k

#

+

,

where A_r(t)is the r×r matrix whose(i,j)-entry is

min(m−i,n−j) k=0

∑

m−i k

n−j k

t^k. Proof. By applyingnmtimes Lemma 14, we obtain that

wHSJg_r(t) =



(1−t^D)

"

(1−t^D). . .

"

(1−t^D) detA_r(t^D)

t^D(^r2)(1−t)^k(1−t^D)^(n+m−r)r

#

+

. . .

#

+





+

.

LetS=∑0≤iaitⁱ∈Z[[t]]be a power series such thata0>0, and leti0∈N∪ {∞}be defined as i₀=

(

∞if for alli≥0,a_i>0;

min({i|a_i≤0})otherwise.

Therefore,[S(t)]₊=∑0≤i<i₀aitⁱ. By convention, fori<0, we putai=0. Then (1−t^D)S(t) = ∑0≤i(ai−ai−D)tⁱ

(1−t^D) [S(t)]₊ = ∑_0≤i<i₀(ai−ai−D)tⁱ .

Consequently, the coefficients of(1−t^D)S(t)and of(1−t^D) [S(t)]₊are equal up to the indexi₀.

• Ifi0=∞, then(1−t^D)S(t) = (1−t^D) [S(t)]₊and hence (1−t^D)S(t)

+=

(1−t^D) [S(t)]₊

+;

• if i₀<∞, then a_i₀−D is positive and thus a_i₀−a_i₀−D is negative. Let i₁ be the index of the first non-positive coefficient of(1−t^D)S(t). Theni₁<i₀, and hence

(1−t^D)S(t)

+= (1−t^D) [S(t)]₊

+.

Therefore, for all power seriesS∈Z[[t]]such thatS(0)>0, we have (1−t^D) [S]₊

+=

(1−t^D)S

+. Consequently, an induction shows that

wHSJg_r(t) =

"

(1−t^D)^{(n−r)(m−r)} detA(t^D) t^D(^r2)(1−t)^k

#

+

. Then, by Corollary 6,wHS

Jg_r(t) =wHS

Ifr(t). Finally, by Lemma 9, we conclude thatHS_I_r(t) = wHSIfr(t). 2

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6. Complexity analysis

Using the previous results on the Hilbert series ofIr, we analyze now the arithmetic complexity of solving the generalized MinRank problem with Gr¨obner bases algorithms. In the first part of this section (until Section 6.2), we consider the homogeneous MinRank problem (i.e. the polynomials f_i,jare homogeneous).

Computing a Gr¨obner basis of the idealϕa(Ir) for the lexicographical ordering yields an explicit description of the set of pointsVsuch that the matrix

ϕa(M) =







ϕa(f_1,1) . . . ϕa(f_1,m) ... . .. ... ϕa(f_n,1) . . . ϕa(f_n,m)







has rank less than r+1. In this section, we study the complexity of this computation when a∈K^nm(^k+D−1D )is generic (i.e.abelongs to a given non-empty Zariski open subset ofK^nm(^k+D−1_D )

) by using the theoretical results from Sections 4 and 5. We focus on the 0-dimensional cases k= (n−r)(m−r)andk<(n−r)(m−r)(over-determined case). Therefore, the set of points where the evaluation of the matrixϕa(M)has rank less thanr+1 is finite.

In order to compute this set of points, we use the following strategy:

• compute a Gr¨obner basis ofϕ_a(Ir)for thegrevlex(graded reverse lexicographical) ordering with theF5algorithm (Faug`ere, 2002);

• convert it into a lexicographical Gröbner basis of ϕa(Ir) by using the FGLM algorithm (Faugère et al., 1993; Faugère and Mou, 2011).

First, we recall some results about the complexity of the algorithmsF₅and FGLM. The two quantities which allow us to estimate their complexity are respectively thedegree of regularity and thedegreeof the ideal. The degree of regularity of a 0-dimensional homogeneous idealI is the smallest integerd such that all monomials of degreed are inI; it is independent on the monomial ordering and it bounds the degrees of the polynomials in a minimal Gr¨obner basis of I. Moreover, in the 0-dimensional case, the Hilbert series is a polynomial from which the degree of regularity can be read off:Dreg(I) =deg(HSI(t)) +1.

In the sequel,ωdenotes a feasible exponent for the matrix multiplication (i.e. a number such that there exists an deterministic algorithm which computes the product of twon×nmatrices inO(n^ω)arithmetic operations inK). The best known bound on this exponent isω <2.3727 (Williams, 2011).

The following proposition and its proof are a variant of a result known in the context of semi- regular sequences (see e.g. Lazard (1983) and Faug`ere (1999) for the relation between Gr¨obner basis computation and linear algebra, Bardet et al. (2004, Proposition 10) and Bardet (2004, Section 3.4) for the complexity analysis).

Proposition 16(Bardet (2004); Bardet et al. (2004)). Let h1, . . . ,h_`∈K[x1, . . . ,x_k] be homogeneous polynomials of degrees d₁, . . . ,d_`, and I=hh₁, . . . ,h_`i. The complexity of computing a Gr¨obner basis of I for a monomial ordering≺is upper bounded by

O

k+Dreg(I) Dreg(I)

−DEG(I) ω−2

k+Dreg(I) Dreg(I)

_`

i=1

∑

k+Dreg(I)−d_i Dreg(I)−d_i

! .