The Improved Mutant Border Basis Algorithm

The experiments with the mutant strategy [142] show that there are two issues. The first arises when the number of lower degree mutants is very large, as this produces many reductions to zero. The second arises when an iteration does not produce mutants at all or produces only an insufficient number of mutants at a lower degree. In later case the mutant strategy makes no improvement. To handle both problems an improved mutant strategy was introduced in [142]. This strategy is based on the mutant concept, however it introduces a heuristic strategy of only choosing the minimum number of mutants, which is called necessary mutants. For details about the improved mutant strategy we refer to Section 3.5. In the following we employ the improved mutant strategy to speed up the Border Basis Algorithm.

Let X = {x1, . . . , xn} be the set of variables, upon which we impose the following order:

x1 > x2 > · · · > xn

Our first goal is to recall a few definitions and concepts to be used later. Recall from Section 2.2, the ring R is the residue class ring of K[x₁, . . . , x_n]. The elements of R are residue classes where each residue class has a unique polynomial representation. Every element f ∈ R \ {0} has a unique representation as a linear combination of terms

f =

s

X

i=0

c_it_i

where ci ∈ K, ti ∈ Tⁿ is a term in R, i.e. ti = x^α₁¹. . . x^α_nⁿ, such that αi < q, 1 ≤ i ≤ n, and where t1 >σ t2 >σ · · · >σ ts. Considering this representation we can define the degree, leading term and leading coefficient in the natural way.

Definition 4.2.8. Let p ∈ R. Then the leading variable of p is the largest variable, according to the order defined on the variables, in the leading term of p and denoted by LV(p).

Definition 4.2.9. Let F_d = {f ∈ F | deg(f ) = d} and let x_i ∈ X. We define F_d^xi as follows.

F_d^xi = {f ∈ F_d | LV(f ) = x_i}

As described above the complexity of the Border Basis Algorithm relies on the computation of a U -stable span. We employ the improved mutant strategy to improve its computation.

Remark 4.2.10. The computation of the U -stable span in Proposition 4.2.2 using the improved mutant strategy can be achieved as follows. Replace step 1) in the algorithm of Proposition 4.2.2 by the following instruction.

1’) Let d_max := max{deg(v) | v ∈ V }, d_min := min{deg(v) | v ∈ V }, P_mutant := ∅, G := V , and let X⁰ := ∅.

Replace step 2) in the algorithm of Proposition 4.2.2 by the following two instructions.

2a) If X⁰ 6= ∅ then choose the smallest variable xl ∈ X⁰ and delete it from X⁰. Otherwise, let X⁰ := {LV(g) | g ∈ G and deg(g) = dmin}, and let delim := dmin+1.

Choose the smallest variable xl ∈ X⁰ and delete it from X⁰. 2b) Let H := ∅. For each g ∈ {g | g ∈ G \ P_mutant and g ∈ G^x_d^l

min} append g to P_mutant and all possible products x_αg with α ∈ {1, . . . , n} to H.

Replace step 5) in the algorithm of Proposition 4.2.2 by the following instruction.

5’) Let H := ∅. If M \ P_mutant 6= ∅, then for each necessary mutant m ∈ M \ P_mutant append m to P_mutant and all possible products x_αm with α ∈ {1, . . . , n} to G.

Let d_elim := min{deg(m) | m ∈ M \ P_mutant} + 1. Then continue with step 3).

The resulting algorithm still computes a vector basis V of the stable span F_U, together with the order ideal U and a K-basis G that contains V of hF i_K. In general, it keeps the size of the computational universe U even smaller than the algorithm in Proposition 4.2.2, but it may require more iterations. In other words, we are sacrificing time efficiency for gaining space efficiency. But experiments with the improved mutant strategy show that it provides space efficiency as well as time efficiency (see [142] and Section 4.3).

Proposition 4.2.11. (The Improved Mutant BBA (MBBA2))

Let F = {f₁, . . . , f_m} ⊂ R be a set of polynomials that generates a zero-dimensional ideal I = hF i in R. Let σ be a degree-compatible term ordering. Consider the following sequence of instructions.

1) Let U be the order ideal generated by Sm

i=1Supp(f_i).

2) Interreduce F to get a K-basis V of hF i_K with pairwise distinct leading terms.

3) Let d_max := max{deg(v) | v ∈ V }, d_min := min{deg(v) | v ∈ V }, X⁰ := ∅, G := V and let P_mutant := ∅.

4) Apply Proposition 4.2.2 along with the modifications given by Remark 4.2.10 to compute a K-basis V of the stable span FU, the updated K-basis G, the updated order ideal U and the updated sets Pmutant and X⁰.

5) Let O = U \ LTσ(V ).

6) If X⁰ 6= ∅ and ∂O * U, then continue with step 4).

7) If ∂O * U , replace U by U⁺, and let d_min := d_max := max{deg(u) | u ∈ U }.

Then continue with step 4).

8) Apply the Final Reduction Algorithm 4.1.7.

This is an algorithm which computes an order ideal O and the O-border basis of I.

Proof. The proof follows from Proposition 4.2.5. The only exception is that in step 4) we compute a K-basis V of F_U in parts. But this does not affect the correctness of the algorithm.

Remark 4.2.12. Assume that we are in the setting of the algorithm in Proposition 4.2.11. Since G, P_mutant, d_min, X⁰ and d_max are initialized by the algorithm itself, we skip step 1’) of the algorithm of Remark 4.2.10 each time it is called to compute a U -stable span. The set of polynomials P_mutant is used to recognize mutants.

Since we are mostly looking for K-rational solutions and our ideal I is a zero-dimensional radical ideal, we can take advantage of the following proposition and theo-rem to choose necessary mutants and to reduce some iterations that produce reductions to zero.

Proposition 4.2.13. Let f1, . . . , fm ∈ P be polynomials which generate a zero-dimensional ideal I = hf1, . . . , fmi. Then the system of equations

f₁(x₁, . . . , x_n) = 0, . . . , f_m(x₁, . . . , x_n) = 0 has at most dimK(P/I) solutions in Kⁿ.

Proof. See [120], Proposition 3.7.5.

The following theorem tells us the exact number of solutions in case the ideal I in Proposition 4.2.13 is a zero-dimensional radical ideal.

Theorem 4.2.14. Let f₁, . . . , f_m ∈ P be polynomials which generate a zero-dimensional radical ideal I = hf₁, . . . , f_mi. Let K be the algebraic closure of K, and let P = K[x₁, . . . , x_n]. If K is a perfect field, the number of solutions of the system of equa-tions

f₁(x₁, . . . , x_n) = 0, . . . , f_m(x₁, . . . , x_n) = 0

is equal to the number of maximal ideals of P containing IP , and this number is precisely dim_K(P/I).

Proof. See [120], Theorem 3.7.19.

In most of the cases we may know the exact number of K-rational solutions, thus we know dim_K(P/I).

Remark 4.2.15. Assume that we are in the setting of Proposition 4.2.11. Since U is a vector space and V ⊆ U is a vector subspace we can form the vector space quotient U/V and we know that

dim_K(U/V ) = dim_KU − dim_KV.

In our case the ideal is a zero-dimensional radical ideal and we may know the exact number of solutions, especially when the system defined by F has a unique solution.

This means that we know dimK(U/V ). Thus in practice, we can check dimKU −dimKV during the course of the algorithm. If at a certain stage the difference dimKU − dimKV equals dimK(U/V ), the U -stable span has been reached and further iterations are producing reductions to zero. This also helps to choose necessary mutants.

Effectiveness of Mutant Strategies for Computing a U -stable span

The complexity of the Border Basis Algorithm relies on the computation of a vector space basis. In Proposition 4.2.2 and Remark 4.2.10 we employed the mutant strategy and the improved mutant strategy respectively to speed up its computation. A natural question could be to ask about the effectiveness of the mutant strategies to compute a vector space basis.

Surprisingly, it turns out that there are deep connections to other mathematical disciplines, and border bases that represent the combinatorial structure of the ideal under consideration in a canonical way. In fact the XL Algorithm (see [59, 117]), which is based on relinearization methods, in its classical form is actually identical to a L-stable span and therefore it is a border basis computation at its core. Recently developed MutantXL [140], MXL2 [142] and MXL3 [143] algorithms are improved versions of the XL Algorithm. This leads us to the following observation.

Assume that we are working over the finite field F2 and using an adapted version of M4RI [4], a library for dense matrix linear algebra over F2. An algorithm to compute a vector space basis using mutant strategies can perform substantially better than the F4 algorithm [73] implemented in Magma [32], currently the best publicly available implementation of F4. This can be observed from experimental results for the Mutan-tXL, the MXL2 and the MXL3 algorithms presented in [37, 142, 143]. Furthermore, a complexity analysis of these algorithms is given in [141], which suggests a new upper bound for the complexity of computing Gr¨obner bases. This complexity analysis also supports our arguments.

Future Work

In the spirit of the algorithms developed in this section, it is obviously possible to generate a number of further variations of the Border Basis Algorithm. Such variations may have the potential to speed up the computation of border bases considerably while at the same time avoiding a quick exhaustion of the available memory. In this sense, the flexible partial enlargement strategy introduced in [36] can definitely improve the computation of a U -stable span. The flexible partial enlargement strategy is a very recent idea and surpasses old boundaries set by mutant strategies.

As a final remark, we point out that by using mutant strategies the algorithms to compute a border basis can compete with the linear algebra algorithms such as F4 [73] and MXL3 [143] which are used for the computation of Gr¨obner bases. Actually, the idea of mutants is more suitable for border bases as compared to Gr¨obner bases.

The preference arises from the iterative generation of linear syzygies, inherent in the border basis algorithm, which allows for successively approximating the basis degree by degree. We point out that the computation of a U -stable span can be found with the help of all the sparse linear algebra techniques which lie at the heart of linear algebra.

Therefore, the Border Basis Algorithm is able to absorb several techniques for speed up and deserves further efficient implementation and experimentation.