On FGLM Algorithms with Tropical Gröbner bases
aa r X i v : . [ c s . S C ] S e p On FGLM Algorithms with Tropical Gröbner bases
Yuki Ishihara
Graduate School of Science, RikkyoUniversityTokyo, [email protected]
Tristan Vaccon
Université de Limoges; CNRS, XLIMUMR 7252Limoges, [email protected]
Kazuhiro Yokoyama
Departement of Mathematics, RikkyoUniversityTokyo, [email protected]
ABSTRACT
Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gröbner bases taking into accountthe valuation of K . Because of the use of the valuation, the theoryof tropical Gröbner bases has proved to provide settings for com-putations over polynomial rings over a p -adic field that are morestable than that of classical Gröbner bases. In this article, we inves-tigate how the FGLM change of ordering algorithm can be adaptedto the tropical setting.As the valuations of the polynomial coefficients are taken intoaccount, the classical FGLM algorithm’s incremental way, monomo-mial by monomial, to compute the multiplication matrices and thechange of basis matrix can not be transposed at all to the tropicalsetting. We mitigate this issue by developing new linear algebraalgorithms and apply them to our new tropical FGLM algorithms.Motivations are twofold. Firstly, to compute tropical varieties,one usually goes through the computation of many tropical Gröb-ner bases defined for varying weights (and then varying term or-ders). For an ideal of dimension 0, the tropical FGLM algorithmprovides an efficient way to go from a tropical Gröbner basis fromone weight to one for another weight. Secondly, the FGLM strat-egy can be applied to go from a tropical Gröbner basis to a classicalGröbner basis. We provide tools to chain the stable computation ofa tropical Gröbner basis (for weight [ , . . . , ] ) with the p -adic sta-bilized variants of FGLM of [RV16] to compute a lexicographicalor shape position basis.All our algorithms have been implemented into SageMath . Weprovide numerical examples to illustrate time-complexity. We thenillustrate the superiority of our strategy regarding to the stabilityof p -adic numerical computations. CCS CONCEPTS • Computing methodologies → Algebraic algorithms . KEYWORDS
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ISSAC ’20, July 20–23, 2020, Kalamata, Greece © ACM Reference Format:
Yuki
Ishihara, Tristan Vaccon, and Kazuhiro Yokoyama. 2020. On FGLM Al-gorithms with Tropical Gröbner bases. In
International Symposium on Sym-bolic and Algebraic Computation (ISSAC ’20), July 20–23, 2020, Kalamata,Greece.
ACM, New York, NY, USA, 9 pages. https://doi.org/10.1145/3373207.3404037
The development of tropical geometry is now more than threedecades old. It has generated significant applications to very vari-ous domains, from algebraic geometry to combinatorics, computerscience, economics, optimisation, non-archimedean geometry andmany more. We refer to [MS15] for a complete introduction.Effective computation of tropical varieties are now available us-ing Gfan and Singular (see [JRS19] , [GRZ19]). Those computa-tions often rely on the computation of so-called tropical Gröbnerbases (we use GB for Gröbner bases in the following). Since Chanand Maclagan’s definition of tropical Gröbner bases taking intoaccount the valuation in [CM19], computations of tropical GB areavailable over fields with trivial or non-trivial valuation, using var-ious methods: Matrix F5 in [Va15], F5 in [VY17, VVY18] or liftingin [MR19].An important motivation for studying the computation of trop-ical GB is their numerical stability. It has been proved in [Va15]that for polynomial ideals over a p -adic field, computing tropicalGB (which by definition take into account the valuation), can besignificantly more stable than classical GB.Unfortunately, no tropical term ordering can be an eliminationorder, hence tropical GB can not be used directly for solving poly-nomial systems. Our work is then motivated by the following ques-tion: can we take advantage of the numerical stability of the com-putation of tropical GB to compute a shape position basis in dimen-sion zero through a change of ordering algorithm?In this article, we tackle this problem by studying the main changeof ordering algorithm, FGLM [FGLM93]. On the way, we inves-tigate some adaptations and optimizations of this algorithm de-signed to take advantage of some special properties of the ideal( e.g. Borel-fixedness of its initial ideal).We also provide a way to go from a tropical term order to an-other. This produces another motivation: difficulty of computationcan vary significantly depending on the term order (see §8.1 of[VVY18]), hence, using a tropical FGLM algorithm, one could gofrom an easy term order to a harder one in an efficient way.Finally, we conclude with numerical data to estimate the lossin precision for the computation of a lex Gröbner basis using atropical F5 algorithm followed by an FGLM algorithm, in an affinesetting, and also numerical data to illustrate the behavior of thevarious variants of FGLM handled along the way.
SSAC ’20, July 20–23, 2020, Kalamata, Greece Yuki Ishihara, Tristan Vaccon, and Kazuhiro Yokoyama
Chan and Maclagan have developed in [CM19] a Buchberger algo-rithm to compute tropical GB for homogeneous input polynomials(using a special division algorithm). Following their work, adapta-tions of the F5 strategies have been developped in [Va15, VY17,VVY18] culminating with complete F5 algorithms for affine inputpolynomials.A completely different approach has been developped by Mark-wig and Ren in [MR19], relating the computation of tropical GB in K [ X , . . . , X n ] to the computation of standard basis in R J t K [ X , . . . , X n ] (for R a subring of the ring of integers of K ). It can be connected tothe Gfanlib interface in Singular to compute tropical varieties (see:[JRS19]).Finally, Görlach, Ren and Zhang have developped in [GRZ19]a way to compute zero-dimensional tropical varieties using shapeposition bases and projections. Their algorithms take as input alex Gröbner basis in shape position. Our strategies can be used toprovide such a basis stably (precision-wise) when working with p -adic numbers, and be chained with their algorithms. Let K be a field with a discrete valuation val such that K is completewith respect to the norm defined by val. We denote by R = O K its ring of integers, m K its maximal ideal (with π a uniformizer),and k = O K / m K its fraction field. We refer to Serre’s Local Fields[Ser79] for an introduction to such fields. Classical examples ofsuch fields are K = Q p , with p -adic valuation, and Q (( X )) or F q (( X )) with X -adic valuation.The polynomial ring K [ X , . . . , X n ] (for some n ∈ Z > ) willbe denoted by A , and for u = ( u , . . . , u n ) ∈ Z n ≥ , we write x u for X u . . . X u n n . For д ∈ A , | д | denotes the total degree of д and A ≤ d the set of all polynomials in A of total degree less than d . Thematrix of a finite list of polynomials (of total degree ≤ d for some d ) written in a basis of monomials (of total degree ≤ d ) is called a Macaulay matrix .For w ∈ Im ( val ) n ⊂ R n and ≤ m a monomial order on A , wedefine ≤ a tropical term order as in the following definition: Definition 1.1.
Given a , b ∈ K ∗ = K \ { } and x α and x β twomonomials in A , we write ax α < bx β if: • | x α | < | x β | , or • | x α | = | x β | , and val ( a ) + w · α > val ( b ) + w · β , or • | x α | = | x β | , val ( a ) + w · α = val ( b ) + w · β and x α < m x β . For u of valuation 0 , we write ax α = ≤ uax α . Accordingly, ax α ≤ bx β if ax α < bx β or ax α = ≤ bx β . Leading terms ( LT ) and leading monomials ( LM ) are defined ac-cording to this term order. See Subsec. 2.3 of [VVY18] for moreinformation on this definition and its comparison with Def. 2.3 of[CM19].Let I ⊂ A be a 0- dimensional . Let B ≤ the canonical linear K -basisof A / I made of the x α < LM ≤ ( I ) . Let δ be the cardinality of B ≤ . We denote by B ≤ the border of B ≤ ( i.e. the x k x α for k ∈ J , n K such that x α ∈ B ≤ and x k x α not in B ≤ ). N F ≤ is the normal formmapping defined by I and ≤ . We define D such that D = + max x α ∈ B ≤ | x α | . The first task in the FGLM strategy is to develop the tools forcomputations in A / I . The main ingredients are the multiplicationmatrices, M , . . . , M n , corresponding to the matrices of the linearmaps given by the multiplication by x i written in the basis B ≤ .Once they are known, it is clear that one can perform any K -algebra operation on elements of A / I written in the basis B ≤ .To compute those matrices, a natural strategy is to go throughthe computation of the normal forms N F ( x i x α ) for x α ∈ B ≤ . We investigate in this section how to proceed with this task, andhow it compares to the classical case.
We recall here the tropical row-echelon form algorithm of [Va15]that we use for computing normal forms using linear algebra.
Algorithm 1:
The tropical row-echelon form algorithm input : M , a Macaulay matrix of degree d in A , with n row rows and n col columns, and mon a list of monomialsindexing the columns of M . output : e M , the U of the tropical LUP-form of M e M ← M ; for i = to n row do Find j such that e M [ i , j ] has the greatest term e M [ i , j ] x mon j for ≤ of the row i ; Swap the columns i and j of e M , and the i and j entries of mon ; By pivoting with the i -th row, eliminates the coefficientsof the other rows on the first column; ; Return e M ;We refer the interested reader to [Va15, VVY18]. We illustratethis algorithm with the following example. Example 2.1.
We present the following Macaulay matrices, over Q [ x , y ] with w = ( , ) , and ≤ m be the graded lexicographical or-dering. The second one is the output of the tropical LUP algorithmapplied on the first one. The monomials indexing the columns arewritten on top of the matrix. x x y y x xy y x x x y xy y y − − −
35 0 −
18 .If all four polynomials represented by the matrix belong to someideal I (and assuming that y , y ∈ B ≤ ( I ) ) then we can concludethat N F ≤ ( xy ) = − y and N F ≤ ( x y ) = − y + y . The classical strategy to compute the
N F ≤ m ( x i x α ) ( x α ∈ B ≤ m )when working with a monomial ordering ≤ m , starting with a re-duced GB G , is to set apart the following only three cases possible: n FGLM Algorithms with Tropical Gröbner bases ISSAC ’20, July 20–23, 2020, Kalamata, Greecen FGLM Algorithms with Tropical Gröbner bases ISSAC ’20, July 20–23, 2020, Kalamata, Greece
N F ≤ m ( x i x α ) ( x α ∈ B ≤ m )when working with a monomial ordering ≤ m , starting with a re-duced GB G , is to set apart the following only three cases possible: n FGLM Algorithms with Tropical Gröbner bases ISSAC ’20, July 20–23, 2020, Kalamata, Greecen FGLM Algorithms with Tropical Gröbner bases ISSAC ’20, July 20–23, 2020, Kalamata, Greece (Type 1) x i x α ∈ B ≤ m ; (Type 2) x i x α ∈ LT ( G ) ; (Type 3) x i x α ∈ LT ≤ m ( I ) but neither in B ≤ m nor in LT ( G ) .Type 1 is the easiest, as in this case N F ≤ m ( x i x α ) = x i x α . Type2 is not very difficult either. If for some д ∈ G , LM ( д ) = x i x α , д = x i x α + Í x β ∈ B ≤ m c β x β , then as G is reduced, we get directlythat N F ≤ m ( x i x α ) = − Í x β ∈ B ≤ m c β x β .Type 3 is the trickiest. We assume that we have already com-puted all the N F ( x j x β ) for x j x β < m x i x α . Let x k be the smallest(for ≤ m ) variable dividing x i x α . Then the normal form N F (cid:18) x i x α x k (cid:19) = Õ x β ∈ B ≤ m , x β < m xi xαxk c β x β is already known. As in the previous sum, x β < m x i x α x k , then x k x β < m x i x α , and all the N F ( x k x β ) ’s are also already known.Therefore, we can write N F ( x i x α ) = Õ x β ∈ B ≤ m , x β < m xi xαxk c β N F ( x k x β ) , and N F ( x i x α ) can be obtained from the previous normal forms.It is easy to see that the cost of computation of a normal formin the third case is in O ( δ ) field operations. The other two casesare negligible. As there are O ( nδ ) multiples to consider, the totalcost for the computation of the multiplication matrices is in O ( nδ ) field operations.Unfortunately, this strategy can not be completely generalizedto the tropical context. There is no issue with the first two com-putations. However, there is no straightforward way to adapt thethird one. We illustrate this failure with the following example. Example 2.2.
Over Q [ x , y ] with ≤ defined by w = ( , ) , and ≤ m ,the graded lexicographical ordering, let us take I = h f , f , f , f i with f = x , f = x y + x y + x y + xy , f = x y + x y + x y + xy , f = y + x y + x y + xy . The first monomialsof the third type arrive in degree 7 , namely xy , x y , x y , x y .Due to the fact that we use a tropical term order, f , f , and f allinvolve the monomials x y , x y , xy . In consequence if one wantsto use multiples of the
N F ( x y ) , N F ( x y ) , N F ( y ) , one gets quan-tity involving each three monomials among xy , x y , x y , and x y . They are all intertwined, and the trick we saw previously formonomials of the third type can not be used. To untangle the reduction of monomials of the third type, we canuse linear algebra. We have to proceed degree by degree. Whilemonomials of the first type do not need any special proceeding, weneed to interreduce the reductions of the monomials of the secondand third types. The general strategy is described in Algorithm 2.
Proposition 2.3.
Algorithm 2 is correct, and is in O ( n δ ) fieldoperations over K . Proof.
The essentially different part compared to the classicalcase starts on Line 13. Lines 16 and 18 are crucial. By definition,monomials of the third type are in LT ( I ) . If x α ∈ L can not bewritten as x k x β with x β of type 2 or 3, it means that all its divisorsare in B ≤ . Consequently, it is a minimal generator ot LT ( I ) and is Algorithm 2:
Multiplication matrices computation algorithm input :
A reduced GB G of the ideal I for ≤ , a tropical termordering. output : M , . . . , M n the multiplication matrices of A / I (overthe basis B ≤ ). Using LT ( G ) , compute B ≤ (and δ = ♯ ( B ≤ ) ); Define M , . . . , M n as zero matrices in K δ × δ , their rows andcolumns are indexed by the x α ∈ B ≤ ; Compute L = { x i x α , for i ∈ J , n K and x α ∈ B ≤ } . ; Compute L = L ∩ ( B ≤ ∪ LT ( G )) c ; for x α ∈ L ∩ B ≤ do for i such that x i divides x α do Set M i [ x α , x α x i ] = /* The column indexed by x α x i is zero, excepton its coefficient indexed by x α / x i */ for x α ∈ L ∩ LT ( G ) do Take д ∈ G such that д can be written д = x α + Í x β ∈ B ≤ д x β x β ; for i such that x i divides x α do for x β ∈ B ≤ do Set M i [ x β , x α x i ] = − д x β ; Set M to be a matrix over K with 0 rows and with columnsindexed by L ∪ LT ( G ) ∪ B ≤ . ; for d a degree of a monomial in L (in ascending order) do for x α ∈ L of degree d do Find x i , and д either in G or as a row of M such that LT ( x i д ) = x α ; Stack x i д at the bottom of M ; Using multiples of the form x i д or д , for д either in G oras a row of M , find a complete set of reducers for all themonomials in L ∪ LT ( G ) appearing with a non-zerocoefficient in their column, and stack them at thebottom of M ; Compute the Tropical Row-echelon form of M byAlgorithm 1 and replace M with it ; for x α ∈ L do Take the row s of M with leading coefficient x α . ; for i such that x i divides x α do for x β ∈ B ≤ do Set M i [ x β , x α x i ] = − M [ s , x β ] M [ s , x α ] ; Return M , . . . , M n of type 2, which is a contradiction. Therefore, any monomial of thethird type is a simple multiple of a monomial of type 2 or 3.As in the for loop on Line 14, we proceed by increasing degree,it is an easy induction to prove that such desired x i and д exist.For the complete set of reducers on Line 18, we use the fact thatthe monomials appearing in M all are in B ≤ ∪ L , again by an easyinduction (using the fact that the rows of M in previous degree SSAC ’20, July 20–23, 2020, Kalamata, Greece Yuki Ishihara, Tristan Vaccon, and Kazuhiro Yokoyama are already reduced), and therefore, the complete set of reducerscan be built.The Tropical Row-echelon form computation then produces thedesired normal forms. The correctness is then clear.Regarding to the arithmetic complexity, we should note thatboth rows and columns of M are indexed by monomials in L ∪ B ≤ and there are O ( nδ ) of them. With the row-reduction, the total costis then in O ( n δ ) arithmetic operations. (cid:3) Remark 2.4.
The matrix M is sparse: any row added to the matrixon Line 17 has at most δ + n sparsity ratio for a better complexity? Example 2.5.
Let G = ( y + x , x + ) be a GB for w = [ , ] and grevlex of the ideal it spans in Q [ x , y ] . Then B ≤ = { , x } , L = { x , y , x , xy } and L = { xy } . Only d = M before and after applying Algorithm 1, M and M : x xy
12 1 01 0 4 xy x
11 0 −
80 1 4 ( x ∗) x − x ( y ∗) x x − We can now analyze the loss in precision when applying Algo-rithms 1 and 2. To prevent loss in precision to explode exponen-tially, we replace Line 5 of Algorithm 1 with the following tworows:(1) By pivoting using the ’leading terms’ of the rows j for j > i ,eliminate all the coefficients possible of row i ;(2) By pivoting with row i , eliminate all the coefficients on the i -th column.The first row makes sense because by construction, all the rowsof M have distinct leading terms, and this is kept unchanged dur-ing the pivoting process. Proposition 2.6.
Let us assume that the matrix built on Line 17of Algorithm 2 has coefficients in K known at precision O ( π N ) . Allrows have distinct leading terms, leading coefficient and let us take Ξ be the smallest valuation of a coefficient of this matrix M . We as-sume that Ξ ≤ . Let l = rank ( M ) . We assume that N > − l Ξ . Then, after the application of Algorithm 1 , the coefficients of the ob-tained matrix ˜ M are known at precision O ( π N + l Ξ ) , and the small-est valuation of a coefficient ˜ M is lower-bounded by l Ξ . Proof.
After the reduction of row 1 by the other rows, the small-est valuation on row 1 is lower-bounded by l Ξ and its coefficientsare known at precision at least O ( π N + l Ξ ) . The coefficients of row1 for the columns indexed by L ∪ LT ( G ) are all zeros, except for itsleading coefficient, which is 1 + O ( π N + ( l − ) Ξ ) . After the reductionof the other rows by row 1, on the rows of index >
1, the coeffi-cients for the columns indexed by L ∪ LT ( G ) are of valuation atleast Ξ and known at precision O ( π N + l Ξ ) . The coefficients for thecolumns indexed by B ≤ are of valuation at least l Ξ and known at using the modification presented just above this proposition the same precision. The desired result follows by an easy inductionargument. (cid:3) We then upper-bound the loss in precision for the whole compu-tation of the multiplication matrices. Recall that: D = + max x α ∈ B ≤ | x α | . Proposition 2.7.
Let us assume that the smallest valuation ofa coefficient of G is Ξ and that the coefficients of G are known atprecision O ( π N ) . As G is reduced, we get that Ξ ≤ .Then the coefficients of the matrices M , . . . , M n are of valuationat least ( nδ ) D Ξ , and are known at precision O π N + (cid:16) ( nδ ) D + − ( nδ ) − (cid:17) Ξ ! . Proof.
This is a corollary to the previous proposition. Thereare at most D calls to the previous proposition, with matrices ofranks l , . . . , l D . Consequently, the upper bound on the valuation is l . . . l D Ξ and the precision is in O ( π N + ( l + l l + ··· + l ... l D ) Ξ ) whichis in O ( π N + D ( l ... l D ) Ξ ) As for all i , l i ≤ nδ , we get the desiredbounds. (cid:3) Remark 2.8.
In the very favorable case where G is homogeneousand w = [ , . . . , ] , we get that Ξ =
0, and no loss in precisionis happening. This is unfortunately not the most interesting casefor polynomial system solving. Numerical data in Section 5 willshow that loss in precision remain very reasonnable when using w = [ , . . . , ] even in the affine case. Following Huot’s PhD thesis [Huo13], when Borel-fixedness (seeSubsec. 3.2) or semi-stability properties are satisfied, many arith-metic operations can be avoided during the computation of themultiplication matrices. We begin with semi-stability.
Definition 2.9. I is said to be semi-stable for x n if for all x α suchthat x α ∈ LM ( I ) and x n | x α we have for all k ∈ J , n − K x k x n x α ∈ LM ( I ) . Semi-stability’s application is explained in Proposition 4.15, The-orem 4.16 and Corollary 4.19 of [Huo13] (see also Section 4 of[FGHR14]). We recall the main idea here with its adaptation to thetropical setting:
Proposition 2.10. If I is semi-stable for x n , M n can be read from G and requires no arithmetic operation. Proof.
The proof is the same as that of Theorem 8 of [FGHR14].We prove that L ∩ x n B ≤ = ∅ . Let x n x α ∈ L ∩ x n B ≤ , with x α ∈ B ≤ . Then there is some monomial m and д ∈ G such that LM ( mд ) = x n x α . As x α ∈ B ≤ , we get that x n ∤ m . Since x n x α ∈ L , then | m | ≥ . Let k < n be such that x k | m . Then, by semi-stability for x n , x α = mx k × x k LM ( д ) x n ∈ LM ( I ) , which is a contradiction. (cid:3) Thanks to Proposition 2.10, Algorithm 3 is correct, and its arith-metic cost is given by the following proposition.
Proposition 2.11.
Given a reduced GB G of the ideal I for ≤ , atropical term ordering, and assuming I is semi-stable for x n , then M n can be computed in O ( δ ) arithmetic operations, which are onlycomputing opposites. n FGLM Algorithms with Tropical Gröbner bases ISSAC ’20, July 20–23, 2020, Kalamata, Greecen FGLM Algorithms with Tropical Gröbner bases ISSAC ’20, July 20–23, 2020, Kalamata, Greece
Given a reduced GB G of the ideal I for ≤ , atropical term ordering, and assuming I is semi-stable for x n , then M n can be computed in O ( δ ) arithmetic operations, which are onlycomputing opposites. n FGLM Algorithms with Tropical Gröbner bases ISSAC ’20, July 20–23, 2020, Kalamata, Greecen FGLM Algorithms with Tropical Gröbner bases ISSAC ’20, July 20–23, 2020, Kalamata, Greece Algorithm 3:
Computing M n , when semi-stable for x n input : A reduced GB G of the ideal I for ≤ , a tropical termordering, assuming I is semi-stable for x n output : M n the matrix of the multiplication by x n in A / I Using LT ( G ) , computes B ≤ (and δ = ♯ ( B ≤ ) ); Define M n as a zero matrix in K δ × δ , its rows and columnsare indexed by the x α ∈ B ≤ ; Compute L n = { x n x α , for x α ∈ B ≤ } . ; for x α ∈ L n ∩ B ≤ do Set M n [ x α , x α x n ] = for x α ∈ L n ∩ LT ( G ) do Take д ∈ G such that д can be written д = x α + Í x β ∈ B ≤ д x β x β . for x β ∈ B ≤ do Set M n [ x β , x α x i ] = − д x β ; Return M n ;To apply the previous result to compute a GB in shape positionin Subsection 4.2, we need to also compute the N F ( x i ) ’s. The fol-lowing lemma states that this is not costly. Lemma 2.12.
Given a reduced GB G of the ideal I for ≤ , a tropicalterm ordering, then the N F ≤ ( x i ) ’s can be computed in O ( nδ ) arith-metic operations, which are only computing opposites. Proof.
It is a consequence of the fact that ≤ is degree-compatible:for any i , x i is either in LT ( G ) or in B ≤ . (cid:3) Subsection 4.2 will apply the previous two results to obtain afast algorithm to compute a shape-position basis.
Remark 2.13.
For grevlex in the classical case, it is known that af-ter a generic change of variable, I is semi-stable for x n . The reasonis that after a generic change of variable, LT ( I ) is equal to the GINof I (see Definition 4.1.3 of [HH11]) , which is known to be Borel-fixed, and Borel-fixedness implies semi-stability for x n . In Section3, we investigate whether this strategy is still valid in the tropicalcase.
In this section, we introduce the tropical generic initial ideal of a0-dimensional ideal analogously to the classical case, and study itsproperties of Borel-fixedness and semi-stability. The desired goalis to be able to use the fast Algorithm 3 after a (generic) change ofvariable.
We follow the lines of Chapter 4 of [HH11], and use the usual ac-tion of GL n ( K ) on A : ( η , f ( x )) ∈ GL n ( K ) × A η ( f ) : = f ( η ⊤ · x ) . Definition 3.1.
An external product of monomials x α ∧· · ·∧ x α k is called a standard exterior monomial if x α ≥ · · · ≥ x a k . If itsmonomial is standard, a term cx α ∧ · · · ∧ x α k is called a standardexterior term . We define an ordering on standard exterior terms bysetting that: cx α ∧ · · · ∧ x α k ≥ dx β ∧ · · · ∧ x β k if val ( c ) + Í ki = w · α i < val ( d ) + Í ki = w · β i , or val ( c ) + Í ki = w · α i = val ( d ) + Í ki = w · β i and there exists 1 ≤ j ≤ k s.t. x α j > x β j and x α i = x β i forall i < j . We then define the leading term of an external product of polynomials f ∧ · · · ∧ f k as its largest term, and denote it by LT ( f ∧ · · · ∧ f k ) . The monomial of the leading term is denoted by LM ( f ∧ · · · ∧ f k ) . Lemma 3.2.
Let ( f , . . . , f t ) ∈ A t . If LT ( f ) > · · · > LT ( f t ) , then LT ( f ∧ · · · ∧ f t ) = LT ( f ) ∧ · · · ∧ LT ( f t ) . Proof.
Let c i be the coefficient of LM ( f i ) in f i . Then, c = Î c i is the coefficient of Γ = LT ( f )∧· · ·∧ LT ( f t ) in f ∧· · ·∧ f t . We mayassume that the f i ’s are ordered such that cLT ( f ) ∧ · · · ∧ LT ( f t ) is a standard exterior term. Let ∆ = dv ∧ · · · ∧ v t be anotherterm in f ∧ · · · ∧ f t and d i the coefficient of v i in f i . Let x α i = LM ( f i ) and x β i = v i . Since c i x α i is the leading term of f i , it followsthat val ( c i ) + w · α i ≤ val ( d i ) + w · β i . Thus, Í ti = ( val ( c i ) + w · α i ) ≤ Í ti = ( val ( d i ) + w · β i ) . As val ( c ) = Í ti = c i and val ( d ) = Í ti = d i , we obtain val ( c ) + Í ki = w · α i ≤ val ( d ) + Í ki = w · β i . If theinequality is strict then Γ is strictly bigger than any permutation ofthe monomials of ∆ such that a standard exterior term is obtained.If equality holds. Then, for all i , val ( c i ) + w · α i = val ( d i ) + w · β i and x α i ≥ x β i . As Γ is a standard exterior term, we deduce thatalso in this case, Γ is strictly bigger than any permutation of themonomials of ∆ such that a standard exterior term is obtained. (cid:3) Lemma 3.3.
Let V ⊂ A be a t -dimensional K -vector space. Let w , . . . , w t be monomials with w > · · · > w t . Then the followingconditions are equivalent.(1) the monomials w , . . . , w t form a K -basis of LT ( V ) ,(2) if ( f , . . . , f t ) is a K -basis of V , then LM ( f ∧ · · · ∧ f t ) = w ∧ · · · ∧ w t ,(3) there exists a K -basis ( f , . . . , f t ) of V s.t. LM ( f ∧ · · · ∧ f t ) = w ∧ · · · ∧ w t . Proof. ( ) ⇒ ( ) : We may assume that the f j ’s are monic and LT ( f ) > · · · > LT ( f t ) . Since LT ( f i ) ∈ LT ( V ) , there is j ( i ) s.t. LT ( f i ) = w j ( i ) . As w > · · · > w t , we obtain j ( i ) = i and LT ( f i ) = w i for all i . By Lemma 3.2, LT ( f ∧· · ·∧ f t ) = LT ( f )∧· · ·∧ LT ( f t ) = w ∧ · · · ∧ w t . ( ) ⇒ ( ) : It is obvious by choosing a K -basis f , . . . , f t of V . ( ) ⇒ ( ) : Since dim ( V ) = dim ( LT ( V )) and w , . . . , w t is linearindependent, it is enough to show that w i ∈ LT ( V ) . Let f , . . . , f t be monic polynomials forming a K -basis of V with LT ( f ) > · · · > LT ( f t ) and LT ( f ∧· · ·∧ f t ) = w ∧· · ·∧ w t . By Lemma 3.2, LT ( f ∧· · · ∧ f t ) = LT ( f ) ∧ · · · ∧ LT ( f t ) and thus w i ∈ LT ( V ) . (cid:3) Proposition 3.4.
Let V ⊂ A d be a t -dimensional K -vector spaceand f , . . . , f t a basis of V . Let cw ∧ · · · ∧ w t be the largest (up tomultiplication by an element of valuation ) standard exterior termof Ó t A ≤ d such that there exists η ∈ GL n ( R ) with LT ( η ( f ) ∧ · · · ∧ η ( f t )) = cw ∧ · · · ∧ w t . Let U V = { η ∈ GL n ( R ) | LT ( η ( f ) ∧ · · · ∧ η ( f t )) = ε × cw ∧ · · · ∧ w t , val ( ε ) = } . Then, U V is open in GL n ( R ) and for any η , υ ∈ U V , LT ( ηV ) = LT ( υV ) . Proof.
As only a finite amount of monomials are possible andval ( R ) is discrete and ≥ U V is well-defined. The valuation beingdiscrete, U V is open: LT ( η ( f ) ∧ · · · ∧ η ( f t )) = ε × cw ∧ · · · ∧ w t amounts to val ( q ( η )) < ν for carefully chosen ν ∈ R and polyno-mial q ∈ Z [ k n × n ] . The last statement follows from Lemma 3.3. (cid:3)
SSAC ’20, July 20–23, 2020, Kalamata, Greece Yuki Ishihara, Tristan Vaccon, and Kazuhiro Yokoyama
From Lemma 3.3, w ∧ · · · ∧ w t in Prop 3.4 is independent of thechoice of basis of V . For d ∈ Z ≥ , let I ≤ d = I ∩ A ≤ d . Theorem 3.5.
Let I be a -dimensional ideal with δ = dim K K [ X ]/ I .We consider the finite dimensional K -vector space I ≤ δ . Then the non-empty open set U I : = U I ≤ δ ⊂ GL n ( R ) satisfies that LT ( ηI ) = LT ( υI ) for any η , υ ∈ U I . Proof.
Let η ∈ U I . We denote LT ( ηI ≤ d ) by J ≤ d . Then J ≤ d = LT ( υI ≤ d ) for all υ ∈ U I and d > δ . Indeed, since LT ( ηI ≤ δ ) containsthe initial terms in the reduced Gröbner basis G of ηI , J ≤ d ⊂ A ≤ d − δ LT ( ηI ≤ δ ) = A ≤ d − δ LT ( υI ≤ δ ) ⊂ LT ( υI ≤ d ) . As dim K ( J d ) = dim K ( LT ( υI d )) , we obtain J d = LT ( υI d ) for all υ ∈ U I . Since LT ( ηI ) = Ð ∞ d = δ J ≤ d , then LT ( ηI ) = LT ( υI ) for any η , υ ∈ U I , which concludes the proof. (cid:3) Definition 3.6.
We call LM ( ηI ) , with η ∈ U I ⊂ GL n ( R ) as givenin Theorem 3.5, the tropical generic initial ideal (tropical gin) of I .Unfortunately, U I is not a Zariski-open subset of GL n ( R ) in gen-eral, hence the generic in the name "tropical gin" is only given as areference to the classical case. The following proposition is a con-solation. Proposition 3.7.
Assume k is infinite. Then U I mod π : = { η mod π , for η ∈ U I } is a non-empty Zariski-open set of GL n ( k ) . Proof.
Let q be the polynomial defining U I ≤ δ in the proof ofTheorem 3.5. One can replace q by some q / π l so that q = q mod π is non-zero, and one can check that consequently, since k is infinite, U I mod π = { x ∈ GL n ( k ) : q ( x ) , } and this is a non-emptyZariski-open set of GL n ( k ) . (cid:3) Remark 3.8.
If, e.g. , R = R J t K , and one takes η ∈ GL n ( R ) at ran-dom using a nonatomic distribution over R , then η belongs to U I with probability one. In classical cases, a generic initial ideal is Borel-fixed ideal i.e. it isfixed under the action of the Borel subgroup
B ⊂ GL n ( K ) , which isthe subgroup of all nonsingular upper triangular matrices. In tropi-cal cases, a generic initial ideal is not always Borel-fixed. However,it can be Borel-fixed under some conditions. Example 3.9.
Let I = ( x , y ) and K = Q (using w = [ , ] andgrevlex). Then in degree two, for a generic change of variables of x ∧ y by the matrix (cid:20) a bc d (cid:21) , we get in K [ x , y ] Ó K [ x , y ] :2 ( a bd − ab c ) x ∧ xy + ( a d − b c ) x ∧ y + ( acd − bc d ) xy ∧ y . Hence the tropical GIN is x ∧ y for degree two, and is thereforenot Borel-fixed, nor semi-stable for y . Definition 3.10.
Let B ⊂ GL n ( O K ) be the subgroup generatedby nonsingular upper triangular matrices whose diagonal entrieshave valuation 0 . We call B a Borel subgroup. We say that a mono-mial ideal J is tropical Borel-fixed if J is fixed under the action of B . A direct adaptation of Theorem 4.2.1 and Prop. 4.2.4 of [HH11]states that the usual properties of the GIN are preserved, undersome conditions. Proposition 3.11.
Let d be the maximal total degree of the re-duced GB of the tropical generic initial ideal of I . If K = Q p and p ≥ d , or if val ( Z \ { }) = { } , then the tropical generic initial idealof I is tropical Borel-fixed and moreover, semi-stable for x n . In this section, we investigate the second part of the FGLM strat-egy. Namely, the multiplication matrices of A / I have been com-puted using the algorithms of Section 2, and we can now performoperations in A / I efficiently.The strategy is then to go through projections in A / I of mono-mials and find linear relations among them. When done carefully,these relations provide polynomials in I , whose leading terms forthe new term order can be read on the monomials defining the rela-tion. When processed in the right order, we can obtain from thesepolynomials a minimal GB of I for our new term order. We first begin with the easiest case of starting from a tropical GBand computing a classical GB.It is clear that once the multiplication matrices are obtained,we can directly apply the classical FGLM algorithm (namely Al-gorithm 4.1 of [FGLM93], see also Algorithm 8 of [Huo13]), or its p -adic stabilized version: Algorithm 3 of [RV16]. This part is in O ( nδ ) arithmetic operations. We refer to Prop 3.6 of loc. cit. andobtain the following propositions. Proposition 4.1.
The total complexity to compute a classical GBof I starting from a tropical GB is in O ( n δ ) arithmetic operations. Behavior regarding to precision can be stated the following way.
Proposition 4.2.
Let ≤ be a tropical term ordering and ≤ be amonomial ordering. Let G be an approximate reduced tropical GB for ≤ of the ideal I , with coefficients known up to precision O ( π N ) . Let Ξ be the smallest valuation of a coefficient in G . Let B ≤ and B ≤ bethe canonical bases of A / I for ≤ and ≤ . Let M be the matrix whosecolumns are the N F ≤ ( x β ) for x β ∈ B ≤ . Let cond ≤ , ≤ ( I ) be thebiggest valuation of an invariant factor in the Smith Normal Form of M . Recall that D = + max x α ∈ B ≤ | x α | .Then if N > cond ≤ , ≤ ( I ) − (cid:16) ( nδ ) D + − ( nδ ) − (cid:17) Ξ , we can chain Algo-rithm 2 and Algorithm 3 of [RV16] to obtain an approximate GB G of I for ≤ . The coefficients of the polynomials of G are known up toprecision O π N + (cid:16) ( nδ ) D + − ( nδ ) − (cid:17) Ξ − cond ≤ , ≤ ( I ) ! . We can apply any classical FGLM algorithm if K is an exact field,or a stabilized variant using Smith Normal Form, as in Algorithm6 of [RV16]. We refer to Prop. 4.5 of loc. cit. . Complexity is veryfavorable when we have the combination of Borel-fixedness andshape position. n FGLM Algorithms with Tropical Gröbner bases ISSAC ’20, July 20–23, 2020, Kalamata, Greecen FGLM Algorithms with Tropical Gröbner bases ISSAC ’20, July 20–23, 2020, Kalamata, Greece
Let ≤ be a tropical term ordering and ≤ be amonomial ordering. Let G be an approximate reduced tropical GB for ≤ of the ideal I , with coefficients known up to precision O ( π N ) . Let Ξ be the smallest valuation of a coefficient in G . Let B ≤ and B ≤ bethe canonical bases of A / I for ≤ and ≤ . Let M be the matrix whosecolumns are the N F ≤ ( x β ) for x β ∈ B ≤ . Let cond ≤ , ≤ ( I ) be thebiggest valuation of an invariant factor in the Smith Normal Form of M . Recall that D = + max x α ∈ B ≤ | x α | .Then if N > cond ≤ , ≤ ( I ) − (cid:16) ( nδ ) D + − ( nδ ) − (cid:17) Ξ , we can chain Algo-rithm 2 and Algorithm 3 of [RV16] to obtain an approximate GB G of I for ≤ . The coefficients of the polynomials of G are known up toprecision O π N + (cid:16) ( nδ ) D + − ( nδ ) − (cid:17) Ξ − cond ≤ , ≤ ( I ) ! . We can apply any classical FGLM algorithm if K is an exact field,or a stabilized variant using Smith Normal Form, as in Algorithm6 of [RV16]. We refer to Prop. 4.5 of loc. cit. . Complexity is veryfavorable when we have the combination of Borel-fixedness andshape position. n FGLM Algorithms with Tropical Gröbner bases ISSAC ’20, July 20–23, 2020, Kalamata, Greecen FGLM Algorithms with Tropical Gröbner bases ISSAC ’20, July 20–23, 2020, Kalamata, Greece Proposition 4.3. If I is in shape position and semi-stable for x n ,then we can combine Algorithm 3 with Algorithm 6 of [RV16]). Thetime-complexity is in O ( nδ ) + O ( δ ) arithmetic operations. Proposition 4.4.
Let G be an approximate reduced GB of I , withcoefficients known at precision O ( π N ) . Let Ξ be the smallest valua-tion of a coefficient in G . If ≤ is lex, and if we assume that theideal I is in shape position and LM ≤ ( I ) is semi-stable for x n , thenthe adapted FGLM in Algorithm 6 of [RV16]), computes an approx-imate GB G of I for lex, in shape position. The coefficients of thepolynomials of G are known up to precision O ( π N − cond ≤ , ≤ + δ Ξ ) .Moreover, we can read on M whether the precision was enough ornot, and hence prove after the computation that the result is indeedan approximate GB. We conclude our series of algorithms with a new algorithm to com-pute a tropical GB of I of dimension 0 knowing the multiplicationmatrices of A / I . In the classical case, the vanilla FGLM algorithm goes throughthe monomials x α in ascending order for ≤ , test whether x α is inthe vector space generated (in A / I ) by the monomials x β such that x β < x α , and if so, produce a polynomial in the GB in construc-tion from the relation obtained by this linear relation.In the tropical case, because of the fact that coefficients have tobe taken into account, a relation (in A / I ) between x α and somemonomials x β such that x β < x α is not enough to ensure that x α ∈ LT ≤ ( I ) . We deal with this issue by (1) taking all monomi-als of a given degree at the same time, in a big Macaulay matrix,and (2) reducing them with a special column-reduction algorithmso as to preserve the leading terms.The linear algebra algorithm is presented in Algorithm 5, withthe general tropical FGLM algorithm in Algorithm 4.The fact that Algorithm 5 computes a column-echelon form ofthe matrix (up to column-swapping) along with the pivoting ma-trix is clear. What is left to prove is the compatibility of the pivotingprocess with the computation of the normal forms and the leadingterms according to ≤ . It relies on the following loop-invariant.
Proposition 4.5.
At any point during the execution of Algorithm5, for any x α , the column of M indexed by x α corresponds to thenormal form N F ≤ ( H ) (with respect to ≤ ) of some polynomial H with LT ≤ ( H ) = x α . Proof.
It is true by construction for any column when enter-ing Algorithm 5. Also by construction, all columns are labelled bydistinct monomials. Now let us assume that on Line 4, we are elim-inating a coefficient d on the column labelled by x β using a co-efficient c on the column labelled by x α as pivot. Because of thechoice of pivot on Line 3, we get that c − x α < d − x β . Let usassume that the column indexed by x α corresponds to N F ≤ ( H ) with LT ≤ ( H ) = x α , and the column indexed by x β corresponds to N F ≤ ( Q ) with LT ≤ ( Q ) = x β . Please note that x α , x β . Then af-ter pivoting the second column corresponds to
N F ≤ ( Q − dc − H ) . As LT ≤ ( dc − H ) = dc − x α < x β , the loop-invariant is then pre-served, which is enough to conclude the proof. (cid:3) Theorem 4.6.
Algorithm 4 terminates and is correct: its output isa GB of the ideal I for ≤ . It requires O ( nδ ) arithmetic operations. Algorithm 4:
A tropical FGLM algorithm input : M , . . . , M n the multiplication matrices of A / I , in abasis B ≤ for a tropical term ordering ≤ , a tropicalterm ordering ≤ . output : A GB G of the ideal I for ≤ . L ← { } , G ← ∅ , d ← M ← the matrix with δ rows and 0 columns ; P ← the matrix with 0 rows and 0 columns ; while L , ∅ do Stack on the right of M all the monomials in L of degree d , written in the basis B ≤ using the multiplicationmatrices ; Remove those monomials from L ; Apply Algorithm 5 with M and ≤ , to get a new M andupdate the pivoting matrix P ; /* If M is the matrix of the N F ≤ ( x α ) for x α indexing the columns of M , then M = M P . */ For all the new columns indexed by x α that reduced tozero, add to G the polynomial x α − Í γ , α P γ , α x γ , andremove the multiples of x α from L ; Add to L the x i x α for all i and for all x α new column in M that did not reduce to zero, and remove the duplicates ; d ← d + Return G Algorithm 5:
Column reduction for FGLM input : M a δ × l matrix over K , whose rows and columnsare indexed by monomials. A tropical term ordering ≤ . An invertible s × s matrix P . output : A column-reduction of M compatible with ≤ , anupdated P . if M = then Return M , P ; Find the coefficient M [ i , j ] of row indexed by x β and columnindexed by x α such that M [ i , j ] − x α is smallest, and usingsmallest x β to break ties ; Use this non-zero coefficient to eliminate the othercoefficients on the same row ; Update P accordingly ; Proceed recursively on the remaining rows and columns ; Return M , P Proof.
We use the following loop-invariant: after Line 9 is ex-ecuted, LT ≤ ( G ) contains all the minimal generators in LT ≤ ( I ) ofdegree ≤ d , they each correspond to a reduced-to-zero column of M , and the x β corresponding to non-reduced-to-zero columns of M are all in N S ≤ ( I ) . The proof for this invariant is as follows. As ≤ is degree-compatible, it is clear by linear algebra that rank ( M ) = dim ( A ≤ d / I ≤ d ) . Thanks to Proposition 4.5, the polynomials addedto G are in I , and more precisely, f = x α − Í γ P γ , α x γ as in Line8 is a polynomial such that LT ≤ ( f ) = x α and N F ≤ ( f ) =
0, asgiven in the Proposition. Their LT ≤ ’s are minimal generators of LT ≤ ( I ) by construction (all multiples of previous generators have SSAC ’20, July 20–23, 2020, Kalamata, Greece Yuki Ishihara, Tristan Vaccon, and Kazuhiro Yokoyama been erased). By a dimension argument, no minimal generator ismissing.Once d is big enough for all minimal generators of LT ≤ ( I ) tohave been produced, no monomials can be left in L and the algo-rithm terminates. Termination and correctness are then clear.As columns are labelled by some x i x α with x α ∈ N S ≤ ( I ) thenat most nδ columns are produced in the algorithm. As the rank of M is δ and so is also its number of rows, the column-reduction of agiven column costs O ( δ ) arithmetic operations. Consequently, thetotal cost of the algorithm is in O ( nδ ) arithmetic operations. (cid:3) Remark 4.7.
The previous algorithm remarkably bears the sameasymptotic complexity as the vanilla classical FGLM algorithm ( O ( nδ ) arithmetic operations), regardless of the more involved linear alge-bra part. Could fast linear algebra also be applied here? Example 4.8.
Let ( x + y , y + ) be a GB of the ideal it spans, for w = [ , − ] and grevlex. We compute a GB of the same ideal for w = [ , ] and grevlex. The following matrices are: the polynomialsadded to M (in three batches, by degree), the final state of M andthe final P . In the end, we get ( y + x , x + ) as the output GB.1 x y x − − y − − x y x y − − − P = A toy implementation of our algorithms in
SageMath [Sage] isavailable on https://gist.github.com/TristanVaccon. The followingarrays gather some numerical results. The timings are expressedin seconds of CPU time. For a given p , we take three polynomials with random coefficientsin Z p (using the Haar measure) in Q p [ x , y , z ] of degrees 2 ≤ d ≤ d ≤ d ≤ . D = d + d + d − w = [ , , ] and thegrevlex monomial ordering, and then apply Algorithms 2 and 4to obtain a lex GB. We compare with the strategy of computing aclassical grevlex GB and then applying FGLM to obtain a lex GB.For any given choice of d i ’s, the experiment is repeated 50 times.Coefficients of the initial polynomials are given at high-enoughprecision O ( p N ) for no precision issue to appear (see [RV16] formore on FGLM at finite precision).Coefficients of the output tropical GB or classical GB are knownat individual precision O ( p N − m ) (for some m ∈ Z ) ). We computethe total mean and max on those m ’s on the obtained GB. In thefirst following array, we provide the mean and max for the tropicalstrategy. In the second, to compare classical and tropical, we pro-vide couples for the mean on the 50 ratios of timing per execution( t ), along with the arithmetic ( Σ ) and geometric ( π ) mean of the 50ratios of mean loss in precision per execution. Data for p = Everything was performed on a Ubuntu 16.04 with 2 processors of 2.6GHz and 16GB of RAM.
In average the tropical strategy takes longer, but save a largeamount of precision (for small p ). While the ratio of saved preci-sion may decrease with the degree, the abolute amount of savedprecision is often still very large. We have also noted that the stan-dard deviations for these ratios can be very large. precision (trop.) D = p =
11 103 25 278 60 509 176 1253 300 1783 652 39293 3 21 12 97 36 396 125 634 141 1002 282 2876101 0 1 0 1 1 79 0 2 15 408 0 265519 0 0 0 0 0 0 0 0 0 0 0 0 trop.classical D = t Σ π t Σ π t Σ π t Σ π t Σ π t Σ πp =
20 .4 .3 5 .4 .2 5 .5 .2 5 .6 .2 1.5 .8 .2 9 1 .23 6 .6 .2 6 .5 .2 5 .5 .2 2 .4 .1 1.2 .7 .1 .9 .9 .1
We repeat the same experiments for mean and max loss in pre-cision, but this time we compute a tropical GB for weight w = [ , , ] and then use Algorithm 4 to compute a tropical GB forweight w = [− , , − ] (grevlex for tie-breaks in both cases). Precision-wise, it seems that there is an intrinsic difficulty in computing a lexGB compared to a tropical GB. precision loss D = p = We adapt our setting to Q (( t )) , using entries with coefficients in Z J t K given at precision 50 (using SageMath ’s built-in random func-tion), and apply the ideas of Subsection 2.5 and Section 3. As Q is involved, computations are slow for D ≥ w = [ , , ] + grevlex D = & FGLM) 2.8 9.4 3.9 102 10 1030precision F5 (mean & max) 0 2 0 2 0 3precision FGLM (mean & max) 0 0 0.1 8 0.4 34 REFERENCES [CM19] Chan A., Maclagan D., Gröbner bases over fields with valuations, Math.Comp. 88 (2019), 467-483.[FGHR14] Faugère, J.-C., Gaudry, P., Huot, L., Renault, G., Sub-cubic Change of Or-dering for Gröbner Basis: A Probabilistic Approach, in Proceedings: ISSAC 2014.ACM, Kobe, Japon, pp. 170–177, 2014[FGLM93] Faugère, J.-C., Gianni, P., Lazard, D., Mora, T., Efficient computation ofzero-dimensional Gröbner bases by change of ordering, J. of Symbolic Compu-tation 16 (4), 329–344, 1993[GRZ19] Görlach, P, Ren, Y, Zhang, L., Computing zero-dimensional tropical vari-eties via projections, arXiv:1908.03486[HH11] Herzog J., Hibi T., Monomial Ideals, Springer, 2001[Huo13] Huot, L., Résolution de systèmes polynomiaux et cryptologie sur lescourbes elliptiques, Ph.D. thesis, Université Pierre et Marie Curie (Paris VI),http://tel.archives-ouvertes.fr/tel-00925271[JRS19] Jensen, A., Ren, Y., Schoenemann, H., The gfanlib interface in Singular andits applications, J. of Software for Algebra and Geometry 9 (2019), 81-87[MS15] Maclagan, D. and Sturmfels, B., Introduction to tropical geometry, GraduateStudies in Mathematics, volume 161, AMS, Providence, RI, 2015[MR19] Markwig, T. and Ren, Y., Computing Tropical Varieties Over Fields withValuation, Foundations of Computational Mathematics, 2019[RV16] Renault, G. and Vaccon, T. On the p n FGLM Algorithms with Tropical Gröbner bases ISSAC ’20, July 20–23, 2020, Kalamata, Greecen FGLM Algorithms with Tropical Gröbner bases ISSAC ’20, July 20–23, 2020, Kalamata, Greece