On FGLM Algorithms with Tate Algebras
aa r X i v : . [ c s . S C ] F e b On FGLM Algorithms with Tate Algebras
Xavier Caruso
UniversitΓ© de Bordeaux, CNRS,INRIABordeaux, [email protected]
Tristan Vaccon
UniversitΓ© de Limoges; CNRS, XLIMUMR 7252Limoges, [email protected]
Thibaut Verron
Johannes Kepler University, Institutefor AlgebraLinz, [email protected]
ABSTRACT
Tate introduced in [Ta71] the notion of Tate algebras to serve, inthe context of analytic geometry over the π -adics, as a counterpartof polynomial algebras in classical algebraic geometry. In [CVV19,CVV20] the formalism of GrΓΆbner bases over Tate algebras hasbeen introduced and advanced signature-based algorithms havebeen proposed. In the present article, we extend the FGLM algo-rithm of [FGLM93] to Tate algebras. Beyond allowing for fast changeof ordering, this strategy has two other important beneο¬ts. First, itprovides an eο¬cient algorithm for changing the radii of conver-gence which, in particular, makes eο¬ective the bridge between thepolynomial setting and the Tate setting and may help in speedingup the computation of GrΓΆbner basis over Tate algebras. Second,it gives the foundations for designing a fast algorithm for interre-duction, which could serve as basic primitive in our previous algo-rithms and accelerate them signiο¬cantly. CCS CONCEPTS β’ Computing methodologies β Algebraic algorithms . KEYWORDS
Algorithms, GrΓΆbner bases, Tate algebra, FGLM algorithm, π -adicprecision ACM Reference Format:
Xavier Caruso, Tristan Vaccon, and Thibaut Verron. 2021. On FGLM Algo-rithms with Tate Algebras. In . ACM, New York, NY, USA, 9 pages.
Lying at the intersection of geometry and number theory, one ο¬nds π -adic geometry. A paramount part of this theory is the study of π -adic analytic varieties, ο¬rst deο¬ned by Tate in [Ta71] (see also[FP04]). They have played a key role in many developments ofnumber theory ( e.g. π -adic cohomologies [LS07], π -adic modularforms [Go88]). The main algebraic objects upon which Tateβs ge-ometry is built are Tate algebras and their ideals, formed of conver-gent multivariate power series over a complete discrete valuationο¬eld πΎ ( e.g. πΎ = Q π ). This work was supported by the ANR project CLapβCLap (ANR-18-CE40-0026-01).T. Verron was supported by the Austrian FWF grant P31571-N32.Permission to make digital or hard copies of part or all of this work for personal orclassroom use is granted without fee provided that copies are not made or distributedfor proο¬t or commercial advantage and that copies bear this notice and the full citationon the ο¬rst page. Copyrights for third-party components of this work must be honored.For all other uses, contact the owner/author(s).
Conferenceβ21, July 2021, Washington, DC, USA Β© 2021 Copyright held by the owner/author(s).
In earlier papers [CVV19, CVV20], the authors showed that it ispossible to deο¬ne and compute GrΓΆbner bases of Tate ideals withcoeο¬cients in Z π or Q π , and that the deο¬nitions are compatiblewith the usual theory on polynomials over the residue ο¬eld F π orover the coeο¬cient ring. A major limitation of the algorithms is theincreasing cost of reductions as the precision grows. Our previouspaper [CVV20] addresses the case of expensive reductions to zero,through the use of signature algorithms, but computing the resultof non-trivial reductions remains expensive. Another question leftopen was whether it is possible to exploit overconvergence prop-erties, namely the knowledge that the series we are working withsatisfy a stronger convergence condition.In the present paper, we adapt the classical FGLM algorithmto the case of Tate series, and we show that it gives answers toboth questions, in the case of zero-dimensional ideals. Precisely,we prove the following theorem. Theorem 1.1.
Let πΎ { X; r } and πΎ { X; u } be two Tate algebraswith πΎ { X; r } β πΎ { X; u } .There exists an algorithm that takes as input a reduced GrΓΆbnerbasis πΊ of a -dimensional ideal πΌ of πΎ { X; r } with respect to a givenmonomial ordering and output a GrΓΆbner basis of the ideal πΌ Β· πΎ { X; u } of πΎ { X; u } for another given monomial ordering.Moreover, if π denotes the number of variables, if πΏ is the dimen-sion of the quotient πΎ { X; r }/ πΌ and if prec is the precision at whichthe result is output, the complexity of this algorithm is: β’ π Λ ( ππΏ prec ) operations in the base ο¬eld πΎ for a general πΎ , β’ π Λ ( ππΏ prec Β· log π ) bit operations when πΎ = Q π . We underline that, although the classical FGLM algorithm onlyconcerns change of ordering, our version also permits to changethe radii of convergence of the underlying Tate algebra (namelythe parameters r and u ), and then provides eο¬cient tools for deal-ing with the aforementioned overconvergence situation. In the ex-treme case where r is inο¬nite, it makes eο¬ective the bridge betweenpolynomials and Tate series, that is between classical algebraic ge-ometry and rigid geometry. On a diο¬erent note, being able to per-form such a change of ordering opens up algorithmic strategiesfor overconvergent series, by giving freedom in the choice of theconvergence radii.An additional important outcome of our algorithm is that it canbe slightly modiο¬ed in order to accept certain nonreduced GrΓΆb-ner bases as input. Hence, in many cases, calling it with the sameradii of convergence and the same ordering as input and output,already performs a nontrivial operation: the interreduction of theinput GrΓΆbner basis. Moreover, it has a controlled complexity and Here πΎ denotes the base ο¬eld and r encodes the radii of convergence of our series;we refer to Β§2.1 for the precise deο¬nitions. onferenceβ21, July 2021, Washington, DC, USA Xavier Caruso, Tristan Vaccon, and Thibaut Verron performs actually very well in practice (contrarily to the naive re-duction algorithm). Since the intermediate interreduction of GrΓΆb-ner bases is often the bottleneck in Buchberger and signature algo-rithms in the Tate setting, using our FGLM algorithm (or an adap-tation of it) at this step could lead to a signiο¬cant speed-up. Strategy and ingredients.
In the classical setting, the key step ofthe FGLM algorithm is to convert back and forth between GrΓΆbnerbases and the so-called multiplication matrices, which are deο¬nedas the multiplication maps by the variables in the quotient space.Performing the change of ordering on those multiplication matri-ces then reduces to basic linear algebra. Still in the classical case,thanks to the structure of normal forms, it can be shown that allsteps can be done in sub-cubic time in the number of solutions.In the Tate setting, GrΓΆbner bases are deο¬ned using a term or-dering, taking into account both a monomial ordering in the usualsense and a weight taking into account the degree of the monomi-als, the valuation of the coeο¬cient and the convergence radius ofthe series in the algebra. It is the reason why we will eventuallybe able to change all these parameters at the same time. However,this feature also implies new diο¬culties.Firstly, in the construction of the multiplication matrices, thestructure of the normal forms does not allow us to read the valuesin one pass. Instead, we prove that an iterative process convergesto the correct value of the matrices, and we show how this processcan be done in diο¬erent ways, including, for some particular baseο¬elds, the option of using relaxed arithmetic [vdH97, BvdHL11],which eventually leads to a signiο¬cant improvement of the eο¬-ciency.Secondly, if the change of ordering incurs a change of conver-gence radii, the size of the quotient algebra might change. We showthat it is possible to recover multiplication matrices over the cor-rect quotient by separating eigenspaces depending on the valua-tion of the eigenvalues. The reconstruction of the ο¬nal GrΓΆbnerbasis is ο¬nally achieved using the classical strategy in the residueο¬eld, and then lifting the basis.
Organization of the article.
In Section 2, we introduce the notationsand discuss some primitives of linear algebras over nonarchime-dian ο¬elds which will be used repeatedly later on. The computationof multiplication matrices is addressed in Section 3. In Section 4, weconsider the question of changing radii of convergence and designour ο¬nal algorithm.
Throughout this article, we consider a ο¬eld πΎ equipped with a dis-crete valuation val for which it is complete. We denote its ring ofintegers by πΎ β¦ and ο¬x a uniformizer π of πΎ . The quotient πΎ β¦ / π is called the residue ο¬eld of πΎ and will be denoted by Β― πΎ in whatfollows. Classical examples of such ο¬elds are πΎ = Q π (equippedwith the π -adic valuation) and π (( π )) (equipped with the π -adicvaluation) for any base ο¬eld π .The complexity statements are given with the usual asymptoticnotations π ( π ) and π Λ ( π ) = π ( π log ( π ) π ) for some π .We will consider two diο¬erent models of complexity: arithmeticcomplexity, counting operations in πΎ or πΎ β¦ , and base complexity,taking into account the precision. In the case of equal characteristic ( i.e. char πΎ = char Β― πΎ ), such as π (( π )) , the base complexity countsoperations in the residue ο¬eld, and the correspondence betweenboth models satistο¬es:(Arithmetic complexity) = π Λ (cid:0) (Base complexity) Β· prec (cid:1) . where prec stands for the working precision. On the contrary, inthe case of mixed characteristic, such as Q π , the base complexitycounts bit operations. When the residue ο¬eld is ο¬nite, the corre-spondence between both models satisο¬es:(Arithmetic complexity) = π Λ (cid:0) (Base complexity) Β· prec Β· log | Β― πΎ | (cid:1) . In order to ο¬x notations, we brieο¬y recall the deο¬nition of Tate alge-bras and the theory of GrΓΆbner bases over them. Let r = ( π , . . . , π π ) β Q π . The Tate algebra πΎ { X; r } is deο¬ned by: πΎ { X; r } : = ( Γ i β N π π i X i s.t. π i β πΎ and val ( π i ) β r Β· i ββββββββ | i |β+β +β ) The tuple r is called the convergence log-radii of the Tate alge-bra. We deο¬ne the Gauss valuation of a term π i X i as val ( π i X i ) = val ( π i ) β r Β· i , and the Gauss valuation of Γ π i X i β πΎ { X; r } as theminimum of the Gauss valuations of its terms. The integral Tatealgebra ring πΎ { X; r } β¦ is the subring of πΎ { X; r } consisting of el-ements with nonnegative valuation. In what follows, when r = ( , . . . , ) , we will simply write πΎ { X } instead of πΎ { X; ( , . . . , )} .We ο¬x a classical monomial order β€ π on the set of monomials X i .Given two terms π X i and π X j (with π, π β πΎ Γ ), we write π X i < π X j if val ( π X i ) > val ( π X j ) , or val ( π X i ) = val ( π X j ) and X i < π X j .The leading term of a Tate series Γ π i X i β πΎ { X; r } is, by deο¬nition,its maximal term.A GrΓΆbner basis of an ideal πΌ of πΎ { X; r } is, by deο¬nition, a fam-ily ( π , . . . , π π ) of elements of πΌ with the property that for all π β πΌ ,there exists an index π β { , . . . , π } such that LT ( π π ) divides LT ( π ) .A GrΓΆbner basis ( π , . . . , π π ) is reduced if given a term π‘ of π π whichis not the leading term, π‘ is not divisible by any LT ( π π ) . The fol-lowing theorem is proved in [CVV19]. Theorem 2.1. (1) Any ideal of πΎ { X; r } admits a GrΓΆbner ba-sis.(2) If r = ( , . . . , ) and πΌ is an ideal of πΎ { X } , a family πΊ = ( π , . . . , π π ) consisting of elements of πΎ { X } with Gauss val-uation is a GrΓΆbner basis of πΌ if and only if its reductionmodulo π is a classical GrΓΆbner basis of the quotient ideal ( πΌ β© πΎ { X; r } β¦ )/ π ( πΌ β© πΎ { X; r } β¦ ) of Β― πΎ [ X ] for β€ π . In the present article, we will be particularly interested in -dimensional ideals. By deο¬nition, πΌ is such an ideal if the quotient πΎ { X; r }/ πΌ is a ο¬nite dimensional πΎ -vector space. If πΌ is a -dimen-sional ideal, the set: π΅ = (cid:8) X i with π β N π and X i β LT ( πΌ ) (cid:9) is ο¬nite and forms a πΎ -basis of πΎ { X; r }/ πΌ . It is called the staircase of πΊ . Moreover, if we are given a GrΓΆbner basis ( π , . . . , π π ) of πΌ , thestaircase π΅ consists of all monomials X i which are not divisible byany LT ( π π ) for π varying in { , . . . , π } . This observation implies inparticular that any reduced GrΓΆbner basis of a -dimensional ideal πΌ consists only of polynomials. n FGLM Algorithms with Tate Algebras Conferenceβ21, July 2021, Washington, DC, USA Algorithm 1:
Big ( π , π ) Input : π β πΎ πΏ Γ πΏ , π β R . Output :
A basis π of Big π ( π ) . π π β charpoly ( π ) ; // use [KV04] or [CRV17] Write π π = π Small ,π ,π π Big ,π ,π ; // use [CRV16] π β π Big ,π ,π ( π ) ; // use [PS73] π β ker π ; // use pnumerical kernel from [KV20, Β§2.2.1] return π It is an understatement to say that the FGLM strategy relies heavilyon linear algebra. In the Tate setting, this assertion is even moretrue and new basic operations in linear algebra, which are speciο¬cto non-archimedean base ο¬elds, will be needed. The aim of thissubsection is to review brieο¬y these operations.
Slope decomposition.
Let π be a ο¬nite πΎ -dimensional vector spaceand let π : π β π be a πΎ -linear mapping. Let π π be the char-acteristic polynomial of π . Given an auxiliary real number π , onecan factor π π as a product π π = π Big ,π ,π Γ π Small ,π ,π where π Big ,π ,π (resp. π Small ,π ,π ) is the factor corresponding to all roots (in an alge-braic closure) of valuation < π (resp. valuation β₯ π ). Moreover, both π Big ,π ,π and π Small ,π ,π have coeο¬cients in πΎ . Letting Big π ( π ) de-note the kernel of π Big ,π ,π ( π ) and Small π ( π ) denote that of π Small ,π ,π ( π ) ,the above factorization corresponds to a decomposition of π as adirect sum π = Big π ( π ) β Small π ( π ) . Computing eο¬ciently this de-composition is a basic task in linear algebra over non-archimedeanο¬elds.In this article, we assume that we are given a routine Big whichtakes as input ( π , π ) and outputs (a basis of) the subspace Big π ( π ) .A naive implementation of the procedure Big is reported in Algo-rithm 1. It has cubic complexity in the dimension of π (which willbe enough for our applications) but has the advantage of being nu-merically stable. πΎ β¦ -modules and saturation. As before, we let π be a ο¬nite dimen-sional πΎ -vector space. We recall basic facts about sub- πΎ β¦ -modulesof π and their algorithmic. If π is equipped with a distinguished ba-sis, one can represent a ο¬nitely generated sub- πΎ β¦ -module of π bythe matrix π whose columns are the generators of πΏ . Performingcolumn reduction, one can always assume that π is under Her-mite normal form. With this additional assumption, it is uniquelydetermined by πΏ .Let πΏ and πΏ be sub- πΎ β¦ -modules of π , represented by the squarematrices π and π respectively. The sum πΏ = πΏ + πΏ is then gen-erated by the columns of the block matrix: π = (cid:0) π | π (cid:1) Computing the Hermite normal form of π , one obtains a canonicalmatrix representating πΏ . The cost of this computation is cubic inthe dimension of π with a naive algorithm.We now assume that we are given a sub- πΎ β¦ -module πΏ β π to-gether with a πΎ -linear endomorphism π : π β π . The saturation of πΏ with respect to π is the sub- πΎ β¦ -module of π deο¬ned by:Sat π ( πΏ ) = πΏ + π ( πΏ ) + π ( πΏ ) + Β· Β· Β· + π π ( πΏ ) + Β· Β· Β· Algorithm 2:
Saturate ( π , πΏ ) Input : a πΎ -linear map π : π β π s.t. π = Small ( π ) ,a ο¬nitely generated πΎ β¦ -module πΏ β π Output :
Sat π ( πΏ ) π β πΏ ; π β π ; π€ β β log dim π β ; for π β J , π€ K do π β π + π ( π ) ; π β π ; return π Lemma 2.2.
We assume that π = Small ( π ) . Then: Sat π ( πΏ ) = πΏ + π ( πΏ ) + π ( πΏ ) + Β· Β· Β· + π dim π β ( πΏ ) . In particular, if πΏ is ο¬nitely generated, then Sat π ( πΏ ) is also. Proof.
The assumption on π implies that the coeο¬cients of π π are all in πΎ β¦ . From Cayley-Hamilton theorem, we deduce that π πΏ isa linear combination with coeο¬cients in πΎ β¦ of the π π βs with π < πΏ .The lemma follows. (cid:3) The routine
Saturate presented in Algorithm 2 computes Sat π ( πΏ ) under the assumption that πΏ is ο¬nitely generated and π = Small ( π ) .Indeed, one checks by induction that after the π -th iteration of theloop, one has π = π π and: π = πΏ + π ( πΏ ) + π ( πΏ ) + Β· Β· Β· + π π β ( πΏ ) . Therefore, when π β₯ πΏ , we ο¬nd π = Sat π ( πΏ ) . The complexity ofAlgorithm 2 is equal to the cost of π ( log πΏ ) Hermite reductions. Ifwe use the naive algorithm for this task, we obtain an algorithm ofarithmetic complexity π Λ ( πΏ ) . Remark 2.3.
For a general πΎ -linear mapping π , one always has:Sat π ( πΏ ) = Big ( π ) + πΏ + π ( πΏ ) + π ( πΏ ) + Β· Β· Β· + π dim π β ( πΏ ) provided that πΏ spans π as a πΎ -vector space. Under this assump-tion, one can then combine Algorithms 1 and 2 to compute thesaturation of πΏ with respect to π even when Small ( π ) ( π . Throughout this section, we ο¬x a tuple r = ( π , . . . , π π ) and con-sider the Tate algebra πΎ { X; r } . We consider in addition a -dimensionalideal πΌ of πΎ { X; r } and assume that we are given a GrΓΆbner basis πΊ = ( π , . . . , π π ) of πΌ .The ο¬rst step in the FGLM algorithm is the computation of thematrices of multiplication by the variables on the quotient πΎ { X; r }/ πΌ (which has ο¬nite dimension by assumption). We recall that a πΎ -basis of πΎ { X; r }/ πΌ is given by the staircase π΅ , which consists ofall monomials π with π β LT ( πΌ ) . We let π π be the matrix of themultiplication by π π with respect to this basis. Observe that the ( π,π ) -entry of π π has valuation at least: π£ π,π = val ( ππ π ) β val ( π ) = val ( π ) β π π β val ( π ) . Our goal is to design an algorithm for computing the π π βs. In or-der to express our complexity estimates, we introduce two impor-tant parameters. This ο¬rst one is the degree of the ideal πΏ = | π΅ | = onferenceβ21, July 2021, Washington, DC, USA Xavier Caruso, Tristan Vaccon, and Thibaut Verron dim πΎ { X; r }/ πΌ . The second one, denoted by π , is the size of theboundary of the staircase deο¬ned as Β― π΅ \ π΅ with Β― π΅ = (cid:8) π π π : π β { , . . . , π } ,π β π΅ (cid:9) . Obviously the cardinality of Β― π΅ is at most ππΏ ; thus π β€ ππΏ as well.The theorem we are going to prove is the following (we refer tothe beginning of Section 2 for the deο¬nition of the arithmetic andbase complexity). Theorem 3.1.
There exists an algorithm that takes as input a re-duced GrΓΆbner basis of πΊ and outputs the multiplication matrices π π with ( π,π ) -entry known at precision π ( π prec + π£ π,π ) for a cost of π ( ππΏ prec ) arithmetic operations.Besides, if the base ο¬eld πΎ is either a Laurent series ο¬eld or Q π , theabove complexity can be lowered to π Λ ( ππΏ prec ) base operations. We will also present an algorithm accepting as input certainnonreduced GrΓΆbner basis. This variant is interesting because, insome cases, it will eventually provide a fast algorithm for interre-ducing GrΓΆbner basis.
Throughout this subsection, we assume that πΊ = ( π , . . . , π π ) is reduced and r = ( , . . . , ) . We will explain later on how theseassumptions can be relaxed. For simplicity, we assume in additionthat the π π βs are all monic ( i.e. the coeο¬cients of their leading termsare ). This hypothesis is of course harmless since renormalizingthe π π βs and making them monic does not aο¬ect the fact that πΊ isa GrΓΆbner basis.Computing the π π βs amounts to computing the normal forms of π modulo πΌ for all π in Β― π΅ . In a classical setting, this can be doneiteratively with linear algebra, by considering the monomials fol-lowing the monomial order. Indeed, for π in π΅ and π β { , . . . , π } ,if π π π β π΅ , either π π π is a leading monomial in πΊ , or there ex-ists π β π΅ such that π π π = π π π , and then NF ( π π π ) = π π NF ( π ) .In the classical setting, the normal form of a monomial π only in-volves monomials in π΅ strictly smaller than π , so π π NF ( π ) onlyinvolves monomials in Β― π΅ strictly smaller than π π π . This allows towrite NF ( π π π ) as a linear combination of already computed nor-mal forms.In the case of Tate term orderings, similarly to what was ob-served for example for tropical orderings [IVY20], the normal formof a monomial π can involve all monomials of π΅ , and computingthe wanted normal forms a priori requires solving a large nonlinearsystem of equations.However, because Tate GrΓΆbner basis are just classical GrΓΆbnerbasis when they are reduced modulo π (Theorem 2.1), the abovestrategy allows to recover the value of the multiplication matricesmodulo π . Following the same computations again lifts the multi-plication matrices to coeο¬cients in πΎ β¦ / π , and so on and so forth.The algorithm formalizing this idea is described in Algorithm 3.For π β πΎ [ X ] with support contained in π΅ , the notation [ π ] rep-resents the vector of coeο¬cients of π in the basis π΅ . With that no-tation, given a monomial π β π΅ and a matrix π with rows andcolumns indexed by π΅ , π Β· [ π ] is the column of π correspondingto π . Algorithm 3:
MulMat_iter ( πΊ, prec ) Input : a reduced GrΓΆbner basis πΊ of the ideal πΌ β πΎ { X } ,an integer prec such that all elements of πΊ are known at precision prec Output : π , . . . ,π π , the multiplication matrices over πΎ { X }/ πΌ (w.r.t the basis π΅ ) modulo π prec π΅ β { π monomials not divisible by any LT ( π ) , π β πΊ } ; π π = ( π π,π,π β² ) β zero matrices of size πΏ Γ πΏ , with rowsand columns indexed by π΅ , for all π β J , π K ; for π from to prec do for π β J , π K , π β π΅ in increasing order of π π π do if π π π β π΅ then π π Β· [ π ] β [ π π π ] ; else if π π π = LT ( π ) for some π β πΊ then π π Β· [ π ] β [ π β LT ( π )] ; else Write π = π π π β² for some π β² β π΅ ; π π Β· [ π ] β π π Β· ( π π Β· [ π β² ]) ; return π , . . . ,π π For π β { , . . . , π } , π,π β π΅ , we denote by π π,π,π the value at row π and column π in the multiplication matrix π π . The following the-orem states the correctness and the complexity of the algorithm. Theorem 3.2.
Algorithm 3 is correct. More precisely, at the endof the π -th run of the loop, the matrices ( π π ) are correct modulo π π .Furthermore, each run through the loop requires π ( πΏ π ) operationsin πΎ β¦ . The proof uses the following observation, which is the transla-tion to the Tate setting of the structure of the normal forms of thestaircase in the classical setting.
Lemma 3.3. If π π π β π΅ , then val ( π π,π,π ) > if π β π π π . Other-wise, if π π π β€ π , then val ( π π,π,π ) > . Proof.
By deο¬nition, the column indexed by π in the multipli-cation matrix π π is the vector of the coordinates of the normalform π of π π π modulo πΊ , in the basis π΅ . If π π π β π΅ , then π = π π π and the result is clear. Otherwise, if π π,π,π π is a term of π , then π π,π,π π < π π π , which, by deο¬nition of the Tate term ordering,means that either π < π π π or val ( π π,π,π ) > . (cid:3) Proof of the theorem.
We prove the result by induction on π β₯ , and, for each value of π , by induction on π π π , π β { , . . . , π } , π β π΅ .The initial case π = is empty. Let π > , π β { , . . . , π } and π β π΅ , and assume by induction that we know the coeο¬cient π π,π β² ,π β² with precision π if π π π β² < π π π , and with precision π β otherwise.If π π π β π΅ , then there is nothing to prove, because the coeο¬-cients are 0 or 1. If π π π = LT ( π ) for some π β πΊ , there is alsonothing to prove, since all the coeο¬cients of π π Β· [ π ] are knownto precision prec β₯ π . n FGLM Algorithms with Tate Algebras Conferenceβ21, July 2021, Washington, DC, USA In the remaining case, for each π β π΅ , the algorithm performsthe substitution π π,π,π β Γ π β² β π΅ π π,π,π β² π π,π β² ,π β² . Since π π π β² = π , π β² < π and π π π β² < π π π and so the inductionhypothesis applies. Let π β² β π΅ . Note that π π π β² β π π π : otherwise, π π π β² = π π π π π β² so π β² = π π π β² , which cannot lie in π΅ . If π π π β² < π π π ,then both π π,π,π β² and π π,π β² ,π β² are known up to precision π (inductionon π π π ), so the product is known to precision π . And if π π π β² > π π π ,then π π,π,π β² is known up to precision π β (induction on π ) and π π,π β² ,π β² is known up to precision π (induction on π π π ) and divisibleby π (by Lemma 3.3), so the product is known up to precision π .For the number of operations, observe that there are less than π pairs ( π,π ) such that the alsorithm needs to perform the compu-tation at line 3.11; each computation involves πΏ coeο¬cients of thematrix, and for each of them, πΏ products in πΎ β¦ . (cid:3) An interesting feature of the algorithm above is that contrary to theusual case, it has to handle monomials which are larger than thecurrent monomial π π π . This removes the main reason for the re-quirement that the input GrΓΆbner basis is reduced, and with slightmodiο¬cations, it can handle any GrΓΆbner basis as long as it is re-duced modulo π . Precisely, this is achieved by replacing line 3.8with the following. Algorithm 3a:
Update the matrices using a nonreducedbasis element for ππ in the support of π β LT ( π ) do if π < π π π then π π Β· [ π ] β π π Β· [ π ] + π [ π ] else Pick π β² β π΅ such that π = π πΌ Β· Β· Β· π πΌ π π π β² ; π π Β· [ π ] β π π Β· [ π ] + ππ πΌ Β· Β· Β· π πΌ π π Β· [ π β² ] ;Unless the staircase is trivial, i.e. as long as the ideal is proper, itis always possible to ο¬nd a suitable π β² at line 3.8e, by picking themonomial β π΅ . Nonetheless, to avoid computing large powers ofmatrices, it is more eο¬cient to ο¬nd π β² as large as possible.It is still true that any monomial π > π π π appearing in theprocess necessarily carries a coeο¬cient with valuation β₯ , andthus the loop invariant that the coeο¬cients are known to precision π still holds.The complexity of the computation is no longer bounded merelyin terms of πΏ , π and prec, but also depends on the degree of thenonreduced terms in the basis, and on the choices of the monomi-als π β² . As described above, the computations can be done in increasingorder of the monomials π π π , ensuring that all the necessary coef-ο¬cients are known with the necessary precision for the next step.Another way to proceed is by dynamic programming, computingthe necessary coeο¬cients recursively if they are not known yet. The recursive deο¬nition, using the matrices π π as a cache, is de-scribed in Algorithm 4, and is very similar to that described in Al-gorithm 3.It can then be called, for all values of π, π, π and π = prec, in-stead of lines 3.3β3.11 in Algorithm 3. The main diο¬erence is thatthe algorithm does not need to specify in which order the coeο¬-cients are computed: the recursive deο¬nition queries the missingcoeο¬cients as needed. The decision on which precision is neededdepends on the valuation of the coeο¬cient: the idea is that if π isknown with precision π and has valuation π£ , and π is known withprecision π and has valuation π€ , then π Β· π is known with precision min ( π + π€, π + π£ ) . Note that it also works if we only know a lowerbound on the valuation, typically if all the digits we know are .The proof that the recursive algorithm terminates is the exis-tence of such an order, as demonstrated in Theorem 3.2. And theproof of complexity is also immediate: there are π coeο¬cients forwhich the calculation is non-trivial, and for each of them, after πΏ multiplications, we gain one digit of precision. The total complex-ity is then π ( ππΏ prec ) operations in πΎ β¦ as in the iterative case.The advantage of the recursive presentation is twofold. Firstly,it will allow in Section 3.4 to generalize the construction, the proofof termination, and the complexity bounds, to arbitrary log-radii.Secondly, it oο¬ers a way to immediately improve the perfor-mance of the algorithms, on coeο¬cient rings such as Z π or π (( π )) where fast arithmetic is available. This works by using a lazy rep-resentation of the number, that is, a representation where eachnumber is the data of its ο¬rst digits, as well as a function allow-ing to compute the next digit. Algorithm 4 gives us precisely sucha function, and as such, the process can be viewed as a recursivedeο¬nition of lazy numbers (the function deο¬nition) together witha delayed evaluation (the function call for all values).For many coeο¬cient rings, it is possible to do better by usingthe so-called relaxed, or on-line, arithmetic. Such arithmetics areavailable for formal power series rings [vdH97] and π -adic num-bers [BvdHL11, BL12]. In that case, the cost of the computationof each new digit (of each variable) is polynomial in log ( prec ) ifwe are counting base operations (in the sense of Section 2). Here,this allows us to compute the matrices with base complexity in π Λ ( ππΏ prec ) . Remark 3.4.
For simplicity and for the complexity bounds, weonly presented the procedure in the case where the GrΓΆbner basisis reduced, but given that the recursive deο¬nition is equivalent tothe loop presented in Algorithm 3, the case where the GrΓΆbnerbasis is not reduced can be dealt with in exactly the same way.
We now consider the case of arbitrary log-radii r β Q π . We willprove that the algorithm presented above still works in that case,by using abstract changes of variables and base ring to justify theexistence of a suitable execution order. Crucially, the algorithmworks without performing those transformations, and the com-plexity is the same. We only need to be more careful about thehandling of the precision and of the valuation.Namely, given r β Q π , we will assume that the input basis πΊ is normalized, in the sense that β€ val ( LT ( π )) < for all π β πΊ . We will further require that for each π β πΊ , and for each π‘ in onferenceβ21, July 2021, Washington, DC, USA Xavier Caruso, Tristan Vaccon, and Thibaut Verron Algorithm 4:
MulMat_rec ( πΊ, π΅, π, π,π, π ) Input : πΊ as in Algo. 3, π΅ the staircase of πΊ , π β J , π K , π β π΅, π β π΅, π β Z , π β€ prec Global : ( π π ) = ( π π,π,π ) π,π β π΅ ) π β J ,π K Output : π π is such that π π,π,π is known to precision π if π π,π,π is known to precision π in π π then do nothing else if π β€ then π π,π,π β π ( ) else if π π π β π΅ then π π,π,π β if π = π π π else else if π π π = LT ( π ) for π β πΊ then π π,π,π β the coordinate of π in the support of π β LT ( π ) else Write π = π π π β² for some π β² β π΅ ; π β ; for π β² β π΅ do π£ β val ( π π,π,π β² ) ; π€ β val ( π π,π β² ,π β² ) ; MulMat_rec ( πΊ, π΅, π, π, π β² , π β π€ ) ; MulMat_rec ( πΊ, π΅, π, π β² ,π β² , π β π£ ) ; π β π + π π,π,π β² π π,π β² ,π β² ; π π,π,π β π + π ( π π + ) ;the support of πΊ , π‘ is known to precision prec + β val ( π‘ )β , and wewill ensure that we compute the matrices with similar precision byensuring that π π,π,π is correct up to precision π + β π£ π,π β .Recall that NF ( π π π ) = Γ π β π΅ π π,π,π π with, for all π , π π,π,π π < π π π . So by deο¬nition of the Tate term ordering, val ( π π,π,π ) β₯ val ( π π π )β val ( π ) = π£ π,π , and the requirement on the precision is merely ad-justing the number of digits we require beyond those we alreadyknow to be 0. The only diο¬erence is that each term is initializedwith the zero digits and the precision which we already know: Algorithm 4a:
Base case with non-zero log-radii else if π β€ β val ( π π π ) β val ( π )β then π π,π,π β π ( π β val ( π π π )β val ( π ) β ) ; Theorem 3.5.
Let r β Q π be a system of log-radii. Algorithm 3,with input a reduced GrΓΆbner basis of an ideal in πΎ { X; r } β¦ , and mod-iο¬ed to compute the matrices using Algorithm 4, computes the mul-tiplication matrices in π ( ππΏ prec ) multiplications in πΎ β¦ . Proof.
Let Ξ ( πΊ ) be the dependency graph of the recurrencerelation deο¬ned in Algorithm 4 with the modiο¬cations of Algo-rithm 4a: namely, Ξ ( πΊ ) is a directed graph whose vertices are tu-ples ( π, π,π, π ) , and there is a directed edge ( π, π, π, π ) β ( π, π β² ,π β² , π ) if and only if the computation of π π,π,π to precision π queries thecoeο¬cient π π,π β² ,π β² to precision π . Note that the vertices with nooutgoing edge correspond to coeο¬cients which are immediatelyknown to precision prec. The recursive computation terminates if and only if the graph is cycle-free, namely, if every path throughthe graph eventually reaches a vertex with no outgoing edge.In the case of trivial log-radii r = ( , . . . , ) , the proof of thatfact is Theorem 3.2. Assume that r β Q π . Let π· be the commondenominator of the log-radii, so that r = ( π / π·, . . . , π π / π· ) . With-out loss of generality, we may assume that πΊ is minimal: elementsof πΊ which can be removed will not take part in the computation.Consider the ο¬eld extension πΏ = πΎ [ π ] with π π· = π , and performthe change of variables π π β π π π π π . This change of variables trans-forms πΊ into a GrΓΆbner basis πΊ β² of an ideal in πΏ β¦ { Y } . In this case,the algorithm terminates, so the graph πΊ β² is cycle-free. If πΊ is mini-mal, so is πΊ β² , and by [CVV19, Prop. 3.10], the elements of this basislie in πΎ β¦ { Y } β πΏ β¦ { Y } . In particular, all throughout the algorithm,the coeο¬cients of the matrices are in πΎ β¦ .The graph Ξ ( πΊ ) is isomorphic to a subgraph of Ξ ( πΊ β² ) , the inclu-sion being given by ( π, π,π, π + β π£ π,π β β ( π, π, π, π ) . Since Ξ ( πΊ β² ) is cycle-free, so is Ξ ( πΊ ) and the algorithm terminates.The bound on the number of operations can be obtained witha similar argument as before, or read on the graph: the complex-ity is bounded by πΏ times the number of vertices of Ξ ( πΊ ) sincecomputing each new vertex has a cost of πΏ operations in πΎ (theadditions and multiplications on line 4.16). Since Ξ ( πΊ ) has at most ππΏ Β· prec vertices, the bound π ( ππΏ prec ) follows. (cid:3) The next step in the FGLM algorithm consists in going in the op-posite direction: starting from multiplication matrices and a termordering, we aim at reconstructing the underlying GrΓΆbner basis.Moreover, in our setting where we want to be able to handle inaddition changes of log-radii, a preliminary step is needed. Indeed,the multiplication matrices are usually aο¬ected by a modiο¬cationof the log-radii. For example, the ideal generated by π₯ β π¦ and π¦ β π₯ in Q [ π₯, π¦ ] has staircase { , π¦, π₯, π¦ , π₯π¦, π₯π¦ } (for lex) whileit spans an ideal over Q { π₯, π¦ } with staircase { , π¦ } (still using lex).We study this phenomenon in full generality in Section 4.1.A toy implementation of the algorithms of this Section is avail-able on https://gist.github.com/TristanVaccon. Theoretical results.
Let r and u be two π -tuples such that π π β₯ π’ π for all π . Under this assumption the Tate algebra πΎ { X; r } is in-cluded in πΎ { X; u } and, given an ideal πΌ in πΎ { X; r } , it makes senseto consider the ideal π½ = πΌ Β· πΎ { X; u } of πΎ { X; u } .In what follows, we always assume that πΌ is -dimensional. Thequotient πΎ { X; r } is then, by deο¬nition, a ο¬nite dimensional πΎ -vectorspace; we will denote it by π . Similarly, we set π = πΎ { X; u }/ π½ .The inclusion πΎ { X; r } β© β πΎ { X; u } induces a πΎ -linear mapping Ξ¦ : π β π .In order to study Ξ¦ , we use topological arguments. We let k Β· k u be the norm on πΎ { X; u } associated to the Gauss valuation val u and equip πΎ { X; u } with the topology associated to this norm. Lemma 4.1.
The ideal π½ is the closure of πΌ in πΎ { X; u } . The results of this section can be extended without diο¬culty to π π = +β , i.e. to πΎ [ X ] . n FGLM Algorithms with Tate Algebras Conferenceβ21, July 2021, Washington, DC, USA Proof.
The polynomial ring πΎ [ X ] is dense in πΎ { X; u } for thenorm k Β· k π’ . Therefore, πΎ { X; r } is dense as well, implying that πΌ isdense in π½ . The fact that π½ is closed follows from [Bo14, Chap. 2,Cor. 8]. (cid:3) The norm k Β· k u induces by restriction a norm on πΎ { X; r } (whichis, of course, diο¬erent from the standard norm k Β· k r on this space)and a mapping k Β· k π : π β R + deο¬ned by: k π₯ k π = inf Λ π₯ k Λ π₯ k u where the inο¬num runs over all Λ π₯ β πΎ { X; r } lifting π₯ . In general, kΒ·k π is not a norm but only a semi-norm, meaning that there mightexist elements π₯ β π for which k π₯ k π = . By deο¬nition, the kernel of k Β· k π is the set of such elements; we denote it by π . It is easilyseen that π is a sub- πΎ -vector space of π . Proposition 4.2.
The map Ξ¦ is surjective and its kernel is π . Proof.
We notice that πΎ { X; u } is the completion of πΎ { X; r } forthe norm k Β· k π’ . Combining this observation with Lemma 4.1, wededuce that π appears as the completion of π with respect to thesemi-norm k Β· k π , which is also the completion of π / π . But, since π / π is ο¬nite dimensional, it is already complete. As a conclusion, π β π / π and the proposition is proved. (cid:3) We now assume that we are given the multiplication matrices π , . . . ,π π over π . We want to relate them to π , or equivalentlyto π . This is the content of the following proposition. Proposition 4.3.
With the above notations, we have: π = π Γ π = Big π’ π ( π π ) (where we recall that the notation Big π’ π was deο¬ned in Β§2.2). Proof.
First of all, we observe that, up to replacing πΎ by πΎ [ π / π· ] for a well-chosen integer π· , we can assume without loss of gener-ality that u is in Z π . Replacing π π by π β π’ π π π and r by r β u , we mayfurther suppose that u = ( , . . . , ) .Let π β { , . . . , π } and let π₯ β Big ( π π ) . By deο¬nition, π₯ is killedby π Big , ,π π ( π π ) . In other words, if π β πΎ { X; r } is a lifting of π₯ , theproduct π Big , ,π π ( π π ) Β· π lies in πΌ . Now, we claim that π Big , ,π π ( π π ) is invertible in πΎ { X } because it is a product of factors of the form π β ( β ππ π ) with val ( π ) > . Consequently, π must be an elementof π½ . By Proposition 4.2, we derive π₯ β π , which proves the inclu-sion Big ( π π ) β π . Since this holds for any π , the β part of theProposition is proved.Set π β² = Γ ππ = Big ( π π ) and π β² = π / π β² . From what we havedone so far, we deduce that the semi-norm k Β· k π on π induces asemi-norm on π β² . The proposition will follow if we can prove that k Β· k π is indeed a norm ( i.e. with trivial kernel) on π β² . In order todo so, we consider the unit ball of π β² , namely: π· β² = (cid:8) π₯ β π β² s.t. k π₯ k π β€ (cid:9) . We want to prove that π· β² does not contain any πΎ -line. For this,we remember that the unit ball of πΎ { X } is exactly the πΎ β¦ -modulegenerated by the monomials X i for i varying in N π . Therefore, π· β² is the smallest πΎ β¦ -module stable under the π π βs and containing theimage of β πΎ { X } in π β² . Keeping in mind in addition that the π π βs Algorithm 5:
NewMulMat ( π , . . . ,π π , π£ ) Input : π , . . . ,π π the multiplication matrices over π , π£ the image of β πΎ { X; r } in π , u β Z π Output : π , . . . , π π the multiplication matrices over theunit ball of π , π€ the image of β πΎ { X; r } in π π β { } ; for π β J , π K do π β π + Big ( π π ) ; π β π / π ; πΏ β π£πΎ β¦ ; for π β J , π K do πΏ β Saturate ( π π , πΏ ) ; return π | πΏ , . . . ,π π | πΏ , π£ mod π ;commute pairwise, we get π· β² = Sat π Sat π Β· Β· Β· Sat π π ( πΏ ) where πΏ is the sub- πΎ β¦ -module of π β² generated by the image of .Besides, on π β² , all the eigenvalues of all the π π βs have nonnega-tive valuation since we have quotiented out all the Big ( π π ) βs. Con-sequently Lemma 2.2 applies and shows that π· β² is ο¬nitely gener-ated. In particular, it contains no πΎ -line, as wanted. (cid:3) Explicit computations.
It is straightforward to turn the previoustheoretical analysis into an actual algorithm that computes thespace π β π / π and the multiplication matrices acting on it. Infact, for later use, it will not be enough to express these matricesin any πΎ -basis of π , but we shall really need a πΎ β¦ -basis of the unitball of π (for the norm k Β· k π introduced before).When u = ( , . . . , ) , Algorithm 5 does the job. In the descrip-tion of this algorithm, we have implicitely assumed that all πΎ -vectorspaces and πΎ β¦ -modules are equipped with distinguished bases, andconsequently used the same notation for a matrix and the endo-mophism it represents. All operations on πΎ β¦ -modules can be han-dled using Hermite normal forms as recalled in Β§2.2; similarly, oper-ations on πΎ -vectors spaces can be done using Smith normal forms,which permits to keep better numerical stability.If πΏ denotes the dimension of π = πΎ { X; r }/ πΌ (which is also thesize of the matrices π π βs), Algorithm 5 requires at most π Λ ( ππΏ ) operations in the base ο¬eld πΎ . The ο¬nal step in the FGLM algorithm is the computation of a GrΓΆb-ner basis from the datum of the multiplication matrices.
Trivial log-radii.
We ο¬rst address the case where u = ( , . . . , ) ,which is covered by Algorithm 6 (page 8). This algorithm uses aroutine FGLMField which takes as input a set of π multiplicationmatrices over a ο¬eld (together with the vector representing themonomial ) and a term ordering and returns the correspondingGrΓΆbner basis. A description of such an algorithm performing thistask can be found in many places in the litterature, for example inthe original article by FaugΓ¨re et al. [FGLM93]. Proposition 4.4.
Algorithm 6 is correct and runs in π ( ππΏ ) arith-metic operations where πΏ denotes the dimension of π . onferenceβ21, July 2021, Washington, DC, USA Xavier Caruso, Tristan Vaccon, and Thibaut Verron Algorithm 6: GB ( π , . . . , π π , π€, β€) Input : π , . . . , π π the multiplication matrices over theunit ball of π , π€ the image of β πΎ { X; r } in π , β€ a monomial ordering Output :
A GrΓΆbner basis πΊ of the ideal π½ β πΎ { X; u } Β― πΊ β FGLMField ( π mod π, . . . , π π mod π, π€ mod π, β€) ; π΅ β { π monomials not divisible by any LM ( π ) , π β Β― πΊ } π = ( π β ,π ) β βB ,π β π΅ β zero matrix; // B denotes the distinguished basis of π we are working with π β , β π€ ; for π β π΅ \{ } by increasing order for β€ do write π = π π π β² with π β { , . . . , π } , π β² β π΅ ; π β ,π β π π Β· π β ,π β² ; // product matrix-vector π = ( π β ,π ) β βB ,π β LM ( Β― πΊ ) β zero matrix; for π β LM ( Β― πΊ ) do write π = π π π with π β { , . . . , π } , π β π΅ ; π β ,π β π π Β· π β ,π ; // product matrix-vector π β π β π ; πΊ β (cid:16) π β Γ π β π΅ π π,π π (cid:17) π β LM ( Β― πΊ ) ; return πΊ Proof.
We recall that we assume u = ( , . . . , ) . Let π½ = πΌ Β· πΎ { X } be as in Section 4.1. We recall that π = πΎ { X }/ π½ by deο¬nition. Let π· denote the unit ball of π . From the facts that the unit ball of πΎ { X } is πΎ { X } β¦ and the norm on π β πΎ { X }/ π½ is the quotientnorm, we deduce that π· β πΎ { X } β¦ / π½ β¦ with π½ β¦ = π½ β© πΎ { X } β¦ . Thereductions modulo π of the π π βs are then the multiplication matri-ces on the quotient Β― π½ = π½ β¦ / π π½ β¦ . The call to FGLMField then returnsa GrΓΆbner basis of the ideal Β― π½ . From Theorem 2.1.(2), we derive thatthe leading terms of a GrΓΆbner basis of π½ are formed by the mono-mials in LM ( Β― πΊ ) . It follows from this that π΅ is the staircase of theideal π½ . In particular, its cardinality is the dimension of π , showingthat the matrix π is a square matrix. After the loops, the columnsof π (resp. of π ) contain the coordinates of the π βs (resp. the π βs)in the distinguished basis B for π varying in π΅ (resp. for π varyingin LT ( πΊ ) ). The matrix π = π β π then contains the expression ofthe π βs in terms of linear combination of the π βs. This shows thecorrectness of the algorithm.The fact that the complexity is in π ( ππΏ ) arithmetic operationsis easily checked. (cid:3) General log-radii.
We now consider the general case where u = ( π’ , . . . , π’ π ) β Q π . We take π· β Z > to be a common denominatorof the coordinates of u (consequently π· Β· u β Z π ). We deο¬ne theο¬eld extension πΏ = πΎ [ π ] such that π π· = π and perform the changeof variables Λ π π = π π·π’ π π π . The Tate algebra πΏ β πΎ πΎ { X; u } becomesisomorphic to πΏ { ΛX } and we can then apply all what precedes with πΏ β πΎ πΎ { X; u } .Inside πΏ { ΛX } β πΏ β πΎ πΎ { X; u } sits the subset π Z πΎ { X; u } consist-ing of series of the form π π£ π with π£ β Z and π β πΎ { X; u } . Let πΌ πΏ , π½ πΏ , π πΏ and π πΏ denote the spaces deduces by πΌ , π½ , π and π re-spectively by extending scalars from πΎ to πΏ . Inside them, we cansimilarly deο¬ne π Z πΌ , π Z π½ , π Z π and π Z π . We claim that then Algo-rithms 5 and 6 can be adapted so that they only have to manipulatevectors lying in these subsets. Indeed: β’ the Big ( Λ π π ) βs can be computed without passing to πΏ becausethey are equal to the Big π’ π ( π π ) βs which are deο¬ned over πΎ ; β’ similarly the quotient π / π is deο¬ned over πΎ and then does notcreate any trouble; β’ one checks that the Hermite normal form of a matrix whosecolumn vectors are in π Z π , remains of this form; β’ the column vectors of the matrices π and π of Algorithm (6)all come from monomials and so have the required shape.Proceeding this way, we avoid the time penalty due to scalar ex-tension from πΎ to πΏ and keep a complexity of π ( ππΏ ) arithmeticoperations. At the end of the day, the output of Algorithm 6 is thena GrΓΆbner basis πΊ of π½ πΏ consisting of series in π Z πΎ { X; u } . Still stay-ing in the same subset, we can normalize these series so that theyall have Gauss valuation . In this case, πΊ is not only a GrΓΆbnerbasis of π½ πΏ but also a GrΓΆbner basis on the ideal π½ β¦ πΏ = πΏ β πΎ π½ β¦ = π½ πΏ β© πΎ { X; u } β¦ . From [CVV19, Proposition 3.10], we deduce that πΊ β© πΎ { X; u } remains a GrΓΆbner basis of π½ β¦ and hence of π½ . Conclusion.
Combining Algorithms 3 (or 4), 5 and 6 and the abovediscussion for covering the case of arbitrary log-radii, we ο¬nallyend up with a complete FGLM algorithm as announced in Theo-rem 1.1 in the introduction. Plugging Algorithm 3a into the ma-chine, we notice that our algorithm can also accept GrΓΆbner baseswhich are nonreduced as soon as they are reduced modulo themaximal ideal. However, in this case, the complexity may growup rapidly, depending on the shape of the input GrΓΆbner basis.
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