Relaxed p-adic Hensel lifting for algebraic systems

Transcription

1 Relaxed -adic Hensel lifting for algebraic systems Jérémy Berthomieu Laboratoire de Mathématiques Université de Versailles Versailles, France Romain Lebreton Laboratoire d Informatique École olytechnique Palaiseau, France lebreton@lix.olytechnique.fr ABSTRACT In a revious article [1], an imlementation of lazy -adic integers with a multilication of quasi-linear comlexity, the so-called relaxed roduct, was resented. Given a ring R and an element in R, we design a relaxed Hensel lifting for algebraic systems from R/ ) to the -adic comletion R of R. Thus, any root of linear and algebraic regular systems can be lifted with a quasi-otimal comlexity. We reort our imlementations in C++ within the comuter algebra system Mathemagix and comare them with Newton oerator. As an alication, we solve linear systems over the integers and comare the running times with Linbox and IML. Keywords Lazy -adic numbers, ower series, algebraic system resolution, relaxed algorithms, comlexity, integer linear systems. 1. INTRODUCTION Let R be an effective commutative ring with unit, which means that algorithms are given for any ring oeration and for zero-testing. Given a roer rincial ideal ) with R, we write R for the comletion of the ring R for the -adic valuation. Any element a R can be written in a non unique way a = i N aii with coefficients a i R. To get a unique writing of elements in R, let us fix a subset M of R such that the rojection π : M R/ ) is a bijection. Then, any element a R can be uniquely written a = i N aii with coefficients a i M. Two classical examles are the comletions k [[X]] of the ring of olynomials k [X] for the ideal X) and Z of the ring of integers Z for the ideal ), with a rime number. In this aer, for R = Z, we take M = {0,..., 1}. Related Works. This aer is the natural extension of [1], which comes in the series of aers [12, 13, 14, 15]. These aers deal with lazy ower series or lazy -adic numbers) Permission to make digital or hard coies of all or art of this work for ersonal or classroom use is granted without fee rovided that coies are not made or distributed for rofit or commercial advantage and that coies bear this notice and the full citation on the first age. To coy otherwise, to reublish, to ost on servers or to redistribute to lists, requires rior secific ermission and/or a fee. ISSAC 12 Grenoble, FRANCE Coyright 20XX ACM X-XXXXX-XX-X/XX/XX...$ and relaxed algorithms. Lazy ower series is the adatation of the lazy evaluation also known as call-by-need) function evaluation scheme for comuter algebra [18]. Lazy ower series are streams of coefficients and the exressions they are involved in are evaluated as soon as the needed coefficients are rovided. It was used in [10] to minimize the number of oerations in that setting. The main drawback of the lazy reresentation is the bad comlexity at high order of some basic algorithms such as multilication. Relaxed algorithms for formal ower series were introduced in [12]. They share with the lazy algorithms the roerty that the coefficients of the outut are comuted one after another and that only minimal knowledge of the inut is required. However, relaxed algorithms differ in the sense that they do not try to minimize the number of oerations of each ste. Since they can anticiate some comutations, they have better comlexity. The first resented relaxed algorithm was for multilication: the so-called online multilication for integers in [6]. Then came the on-line multilication for real numbers in [22], and relaxed multilication for ower series [11, 12], imroved in [13]. One imortant advantage of relaxed algorithms is to allow the comutation of recursive ower series or -adic numbers in a good comlexity for both theoretical and ractical uroses. Another advantage inherited from lazy ower series is that the recision can be increased at any time and the comutation resumes from its revious state. Relaxed algorithms are well-suited when one wants to lift a -adic number to a rational number with no shar a riori estimation on the required recision. On the other hand, there are zealous algorithms. The recision is fixed in advance in the comutations. The Newton- Hensel oerator allows one to solve imlicit equations in R/ n ) for any n N [21]. It has been thoroughly studied and otimized in articular for linear system solving [5, 20, 23]. Our contribution. In this aer, we show how to transform algebraic equations into recursive equations. As a consequence, we can use relaxed algorithms to comute the Hensel lifting of a root from the residue ring R/) to its -adic ring R. We work under the hyothesis of Hensel s lemma, which requires that the derivative at the oint we wish to lift is not zero. Our algorithms are worse by a logarithmic factor in the recision comared to zealous Newton iteration. However, the constant factors hidden in the big-o notation are oten-

2 tially smaller. Moreover, we take advantage of the good evaluation roerties of the imlicit equations. For examle, we found a quasi-otimal algorithm for solving linear systems over -adics see Proosition 21 for s constant). Another examle concerns the multivariate Newton-Hensel oerator which erforms at each ste an evaluation of the imlicit equations and an inversion of its evaluated Jacobian matrix. In Theorems 24 and 26, we manage to save the cost of the inversion of the Jacobian matrix at full recision. Finally, we imlement these algorithms to obtain timings cometitive with Newton and even lower on wide ranges of inut arameters. As an alication, we solve linear systems over the integers and comare to Linbox and IML. We show that we imrove the timings for small matrices and big integers. Our results on the transformation of imlicit equations to recursive equations were discovered indeendently at the same time by [15]. This aer deals with more general recursive ower series defined by algebraic, differential equations or a combination thereof. However, its algorithms have yet to be imlemented and only work in characteristic zero. Furthermore, since the carry is not dealt with, the blockwise roduct as resented in [1, Section 4] cannot be used. This is imortant because blockwise relaxed algorithms are often an efficient alternative. 2. PRELIMINARIES Straight-line rograms. In this aer, we will make use of the straight-line model of comutation to describe the algorithms behind the algebraic and recursive equations. We give a short resentation of this notion and refer to [2] for more details. Let R be a ring and A an R-algebra. A straight-line rogram s.l..) is an ordered sequence of oerations between elements of A. An oeration of arity r is a ma from a subset D of A r to A. We usually work with the binary arithmetic oerations +,, : A 2 A. We also define for r R the 0-ary oerations r c whose outut is the constant r and the unary scalar multilication r by r. We denote the set of all these oerations by R c and R. Finally, let us denote by S the set of regular elements in R, that is of non zero divisors in R. We consider for s S the unary scalar division /s : A S A, and we denote by S their set. Let us fix a set of oerations Ω, usually Ω = {+,, } R S R c. An s.l.. starts with a number l of inut arameters indexed from l 1) to 0. It has k instructions Γ 1,..., Γ k with Γ i = ω i; u i,1,..., u i,ri ) where l < u i,1,..., u i,ri < i and r i is the arity of the oeration ω i Ω. The s.l.. Γ is executable on a = a 0,..., a l 1 ) with result sequence b = b l+1,..., b k ) A l+k, if b i = a l 1+i whenever l 1) i 0 and b i = ω i b u,1,..., b u,ri ) with b u,1,..., b u,ri ) D ωi whenever 1 i k. The multilicative comlexity L Γ) of an s.l.. Γ is the number of oerations ω i that are multilications between elements of A. Examle 1. Let R = Z, A = Z [X, Y ] and Γ be the s.l.. with two inut arameters indexed 1, 0 and Γ 1 = ; 1, 1), Γ 2 = ; 1, 0), Γ 3 = 1 c ), Γ 4 = ; 2, 3), Γ 5 = 3 ; 1). First, its multilicative comlexity is L Γ) = 2. Then, Γ is executable on X, Y ) A 2, and for this inut its result sequence is X, Y, X 2, X 2 Y, 1, X 2 Y 1, 3X 2). Remark 2. For the sake of simlicity, we will associate an arithmetic exression with an s.l.. It is the same oeration as when one writes an arithmetic exression in a rogramming language, e.g. C, and a comiler turns it into an s.l.. In our case, we fix an arbitrary comiler that starts by the left-hand side of an arithmetic exression. For examle, the arithmetic exression ϕ : Z Z 2)2 + 1 can be reresented by the s.l.. with one argument and instructions Γ 1 = ; 0, 0), Γ 2 = ; 1, 1), Γ 3 = 1 c ), Γ 4 = +; 2, 3). Lazy framework and data structures on -adics. We assume that two functions quo and rem are rovided in order to deal with carries that aear when comuting in Z for examle. They are the quotient and the remainder functions by ; quo, ) is a function from R to R and rem, ) is a function from R to M such that for all a in R, a = quoa, ) + rema, ). Then, we give a short resentation of the lazy framework. It states that for any oeration, the outut at a given recision cannot require to know the inut at a recision greater than necessary. For instance, when comuting c n in the roduct c = a b, one cannot make use of a n+1 and b n+1. As a consequence, in the lazy framework, the coefficients of the outut are comuted one by one starting from the term of order 0. Now let s recall the basics of the imlementation of recursive -adic numbers found in [1, 12]. We follow these aers notation and use a C++-style seudo-code. The main class Padic contains the comuted coefficients ϕ : M[] of the number a u till a given order n in N. In addition, any class derived from Padic must contain a method next whose urose is to comute the next coefficient a n. Examle 3. Addition in Z is imlemented in the class Sum Padic derived from Padic and therefore inherits the attributes ϕ : M[] and n. It has additional attributes a, b in Padic and carry γ in M set by default to zero. Finally, the function next adds a n, b n and γ, uts the quotient of the addition in γ and returns the remainder. The class Padic is also endowed with an accessor method [] n N) which calls next ) until the recision reaches n+1 and then oututs ϕ n. Relaxed multilication. Having a relaxed multilication is very convenient for solving recursive algebraic equation in good comlexity. As we will see, solving a recursive equation is very similar to checking it. Therefore, the cost of solving such an equation deends mainly on the cost of evaluating the equation. If, for examle, the equations are sarse, we are able to take advantage of this sarsity. Let a, b R be two -adic numbers. Denote by a n M the coefficients of a in the decomosition a = n N ann. We detail the comutation of the first four terms of c = a b. The coefficients c n are comuted one by one. The major difference with the naive algorithm comes from the use of fast multilication algorithms on significant arts of forthcoming terms. The zeroth coefficient c 0 = rem a 0b 0, ) is comuted normally, a carry γ = quo a 0b 0, ) is then stored. For the first

3 coefficient, one comutes a 0b 1 +a 1b 0 +γ with both roducts a 0b 1 and a 1b 0. Then, the remainder of the division by is assigned to c 1 while the quotient is to γ. Some modifications haen when comuting c 2. By hyothesis, the coefficients a 0, a 1, a 2, b 0, b 1, b 2 M are known. So one can comute a 0b 2 + a 2b 0 + a 1 + a 2) b 1 + b 2) + γ using two multilications of elements of size 1 and one of elements of size 2: a 1 + a 2) b 1 + b 2). Then, c 2 is just the remainder of the division of this number by and γ is the quotient. For the term of order 3, one comutes a 0b 3+a 3b 0+γ, with two multilications of elements of size 1. Instead of the two roducts in a 1b 2+a 2b 1, we used fast algorithms on the bigger data a 1 + a 2 and b 1 + b 2 to save some multilications. Throughout this aer, we measure the cost of an algorithm by the number of arithmetic oerations in R/) it erforms. Notation 1. Let us denote by Mn) the number of arithmetic oerations in R/) needed to multily two elements of R/ n ). In articular, when R = k[x] and = X, it is classical that Mn) On log n log log n) [3]. If R = Z, we rather secify the number of bit-oerations. Two integers of bit-size less than m can be multilied in Im) Om log m 2 log m ) bit-oerations [7], where log is the iterated logarithm. We denote by Rn) the number of arithmetic oerations in R/) necessary to multily two elements of R at recision n. Theorem 4 [1, 6, 12]). The comlexity Rn) for multilying two -adic numbers at recision n, using the relaxed multilication, is O Mn) log n). The revious result was first discovered for integers in [6]. However the alication to comute recursive ower series or -adic integers was seen for the first time in [12]. Article [1] generalizes relaxed algorithms for -adic numbers. Remark 5. If R = F contains many 2 th roots of unity, then two ower series over F) can be multilied in recision n in O M n) e 2 log 2 log log n multilications in F, see [13]. The relaxed algorithm will be used for multilications in R. For divisions in R, we use the relaxed algorithm of [1]. Relaxed recursive -adic numbers. The relaxed model was motivated by its efficient imlementation of recursive - adic numbers. We will work with recursive -adic numbers in a simle case and do not need the general context of recursive -adic numbers [17, Definition 7]. We denote by ν a) the valuation in of the -adic number a. Definition 1. Let Φ R [Y ], y be a fixed oint of Φ, i.e. y = Φ y), and let y 0 = y mod, y 0 M. Let us denote Φ 0 = Id and, for all n N, Φ n = Φ Φ n times). Then, y is a recursive -adic number and Φ allows the comutation of y if, for all n N, we have y Φ n y 0)) n+1 R. The vector case is included in the definition: if denotes,..., ) R r, then R r R r ). Furthermore, the general case with more initial conditions y 0, y 1,..., y l is not considered here but it is believed to be an interesting extension of these results. Proosition 6. Let Φ R [Y ] with a fixed oint y and let y 0 = y mod. Suose ν Φ y 0)) > 0. Then Φ allows the comutation of y. Moreover, for all n m N, Φy)) n does not deend on y m, i.e. Φy)) n = Φy + a)) n for any a n R. Proof. First, notice that y y 0) R. Then, for all y, z R, there exists, by Taylor exansion of Φ at z, a -adic Θy, z) such that Φ y) Φ z) = Φ z) y z) + y z) 2 Θy, z). For all n N, we set y n) := Φ n y 0) and we aly the revious statement to y itself and z = y n) : y y n+1) y y n) = Φ y) Φy n)) y y n) = Φ y n) ) + y y n) )Θy, y n) ). By our assumtion, ν Φ y n) )) > 0, so that we have y y n+1) y y n) R, for all n in N. For the second oint, one has Φy +a) Φy) = Φ y)a+a 2 Θy +a, y) n+1 R since ν Φ y)) > 0. We refer to [1, Section 5.1] for a descrition of the imlementation of a recursive -adic number y with inut Φ and y 0. We recall briefly the ideas to comute y from Φy) in the lazy framework. What we really comute is Φy). Suose that we are at the oint where we know y 0,..., y n 1 and Φy) has been comuted u to its n 1)st coefficient. Since in the lazy framework, the comutation are done ste by ste, one can naturally ask for one more ste of the comutation of Φy). Also, from Proosition 6, Φy)) n deends only on y 0,..., y n 1 so that we can comute it and deduce y n = Φy)) n. But one has to be cautious because, even if Φy) n does not deend on y n, the coefficient y n could still be involved in the comutation of this coefficient. Here is an examle. Warning 7. Take R = Z for any rime number. Let Φ Y ) := Y 2 +, and y be the only fixed oint of Φ satisfying y 0 = 0. Since Φ 0) = 0, Φ allows the comutation of y. At the first ste, we find Φy)) 1 = 1. Then we comute Φy)) 2 = y 2 + ) = y 2). At the second ste of the 2 2 relaxed multilication algorithm, we comute y 0y 2 and q = y 1 + y 2) y 1 + y 2). Here we face a roblem. The -adic number q deends on y 2. Since we do not know y 2 yet, we must roceed otherwise. Shifted algorithms. Because of the issue raised in Warning 7, we need to make the fact that y n is not required in the comutation of Φ y)) n exlicit. This issue was never mentioned in the literature before. In this section, we solve this roblem for olynomial oerators Φ. For this matter, we introduce for all i in N two new oerators: i : R R / i : i R R a i a, a a/ i. Let Ω be the set of oerations { +,,, i, / i} R S R c. In the next definition, we define a number, the shift, that will indicate which coefficients of an argument of an s.l.. are involved in its outut from a syntactic oint of view. Definition 2. Let Γ = Γ 1,..., Γ k ) be an s.l.. over the R-algebra R with l inut arameters and oerations in Ω.

4 For any i, j such that l 1) i k and l 1) j 0, the shift sh Γ, i, j) of its ith result b i with resect to its jth argument is an element of Z {+ }. For i 0, we define sh Γ, i, j) by 0 if i = j and + otherwise. For i > 0, if Γ i = ω i; u, v) with ω i {+,, }, then we set sh Γ, i, j) := min sh Γ, u, j), sh Γ, v, j)); if Γ i = r c ; ), then sh Γ, i, j) := + ; if Γ i = s ; u), then sh Γ, i, j) := sh Γ, u, j) + s; if Γ i = / s ; u), then sh Γ, i, j) := sh Γ, u, j) s; if Γ i = ω; u) with ω R S, then we set sh Γ, i, j) := sh Γ, u, j). We abbreviate sh Γ) := sh Γ, k, 0) if Γ has one argument. With this definition, we can now rove that if an s.l.. Ψ has a ositive shift, then Ψy)) n does not deend on y n. Proosition 8. With the notations of Definition 2, let y = y 0,..., y r 1) R ) r be such that Γ is executable on inut y and b i R be the ith element of the result sequence. Then, for all n N, the comutation of b i) n involves at most the terms y j) l of the argument y j where 0 l max0, n sh Γ, i, j r 1))). Proof. By recursion on the second argument i of sh. When l 1) i 0, the result b i equals to y i+r 1) so that the roosition is easily checked. Now recursively for i > 0. If Γ i = s ; u), then for all n N, b i) n = s b u) n = b u) n s which, by assumtion, deends at most on the coefficients y j) l of the argument y j where 0 l max0, n s sh Γ, i, j r 1))). If Γ i = ; u, v), then for all n N, b i) n = b u b v) n. Since the nth term of the roduct b i deends only on the terms u to n of its argument b u and b v, the roosition follows. The other cases can be treated similarly. Examle 9. We carry on with the notations of Warning 7. Recall that Φ y) n ) involved y l for 0 l n. This is due to the fact that sh Γ) = 0, where Γ is the s.l.. with one argument associated to the arithmetic exression Z Z 2 + see Remark 2). Now consider the s.l.. ) 2 Ψ : Z 2 Z +. Then sh Ψ) = 1, since sh 2 Z/) 2) = sh Z/) 2) +2 = sh Z/) + 2 = sh Z) + 1. So Proosition 8 ensures that the s.l.. Ψ solves the roblem raised in Warning 7. Still, we detail the first stes of the new algorithm to convince even the most sketical reader. At the nth ste, we deduce y n from Ψy)) n = y/) 2 ) n 2 + n. Therefore, Ψy)) 1 = Then Ψy)) 2 = y/) 2 ) and the relaxed roduct algorithm will comute y1. 2 At ste 3, we comute the term 1 of the roduct, which uses y 0y 1. Finally at ste 4, the relaxed roduct algorithm comutes y 1y 3 and q = y 2 + y 3) y 2 + y 3). We have solved the deendency issue. We are now able to exress which s.l.. s Ψ are suited to the imlementation of recursive -adic numbers. Recall that S is the set of regular elements in R. We denote by K := S 1 R the total ring of fractions of R. Definition 3. Let y be a recursive -adic and Φ K [Y ] with denominators not in R that allows the comutation of y. Let Ψ be an s.l.. with one argument and oerations in Ω. Then, Ψ is said to be a shifted algorithm for Φ and y 0 if sh Ψ) 1, Ψ is executable on y, such that y = y 0 mod over the R-algebra R and Ψ comutes Φ Y ) on inut Y over the R-algebra K [Y ]. Next roosition is the cornerstone of comlexity estimates regarding recursive -adic numbers. Proosition 10. Let Ψ be a shifted algorithm for the recursive -adic y whose multilicative comlexity is L. Then, the relaxed -adic y can be comuted at recision n in time L R n) + On). Proof. Proosition 8 roved that we can comute y by comuting Ψy) in the relaxed framework. Therefore the cost of the comutation of y is exactly the cost of the evaluation of Ψ y) in R. We recall that addition in R R, subtraction in R R, multilication in R R that is oerations in R) and division in R S that is oerations in S) u to the recision n can be comuted in time O n). Scalars from R are decomosed in R in constant comlexity. Finally, multilications in R R are done in time R n) see [1]). Now the multilicative comlexity L of Ψ counts exactly the latter oeration. 3. UNIVARIATE ROOT LIFTING In [1, Section 7], it is shown how to comute the dth root of a -adic number a in a recursive relaxed way, d being relatively rime to. In this section, we extend this result to the relaxed lifting of a simle root of any olynomial P R [Y ]. Hensel s lemma ensures that from any modular simle root y 0 R/ ) of P R/ ) [Y ], there exists a unique lifted root y R of P such that y = y 0 mod. From now on, P is a olynomial with coefficients in R and y R is the unique root of P lifted from the modular simle root y 0 R/ ). Proosition 11. The olynomial Φ Y ) := P y 0) Y P Y ) P y 0) allows the comutation of y. K [Y ] Proof. It is clear that if P y) = 0 and P y 0) 0, then y = P y 0 )y P y) P y 0 ) = Φ y). Furthermore, Φ y 0) = 0. In the following subsections, we will derive some shifted algorithms associated to the recursive equation Φ deending on the reresentation of P. 3.1 Dense olynomials In this subsection, we fix a olynomial P of degree d given in dense reresentation, that is as the vector of its coefficients in the monomial basis 1, Y,..., Y d). To have a shifted algorithm, we need to exress Φ Y ) with a ositive shift. Recall, from Definition 2, that the shift of Φ Y ) is 0. ) 2 Lemma 12. The s.l.. Γ : Z 2 Z y0 Z k) for k N {0} is executable on y and sh Γ) = 1.

5 Proof. Since y 0 = y mod, Γy) R and thus Γ is executable on y. Furthermore, ) ) the shift sh Γ) equals 2 + min sh Z y0, sh Z) = 1. We are now able to derive a shifted algorithm for Φ. Algorithm 1 - Dense olynomial root lifting Inut: P R [Y ] with a simle root y 0 in R/ ). Outut: A shifted algorithm Ψ associated to Φ and y Comute Q Y ) the quotient of P Y ) by Y y 0) 2 ) 2 2. Let sq Z) : Z Z y0 3. return the shifted algorithm Ψ : Z 1 P y 0) P y0) P y 0) y Q Z) sq Z)) ). Proosition 13. Given a olynomial P of degree d in dense reresentation and a modular simle root y 0, Algorithm 1 defines a shifted algorithm Ψ associated to Φ. The recomutation of such an oerator involves Od) oerations in R, while we can lift y at recision n in time d 1) R n)+ On). Proof. First, Ψ is a shifted algorithm for Φ. Indeed since sh P y 0) P y 0) y 0) = + and, due to Lemma 12, sh 2 sqz) Q Z)) ) = 1, we have sh Ψ) = 1. Also, thanks to Lemma 12, we can execute Ψ on y over the R- algebra R. Moreover, it is easy to see that Φ Y ) = Ψ Y ) over the R-algebra K [Y ]. The quotient olynomial Q is recomuted in time Od) via the naïve Euclidean division algorithm. Using Horner scheme to evaluate Q Z), we have L Ψ) = d 1 and we can aly Proosition Polynomials as straight-line rograms In [1, Proosition 7.1], the case of the olynomial P Y ) = Y d a was studied. Although the general concet of a shifted algorithm was not introduced, an algorithm of multilicative comlexity O L P )) was given. The shifted algorithm was only resent in the imlementation in Mathemagix [16]. We clarify and generalize this aroach to any olynomial P given as an s.l.. and roose a shifted algorithm Ψ whose comlexity is linear in L P ). In this subsection, we fix a olynomial P given as an s.l.. Γ with oerations in Ω := {+,, } R R c and multilicative comlexity L := L P ), and a modular simle root y 0 R/ ) of P. Then, we define the olynomials T P Y ) := P y 0) + P y 0) Y y 0) and E P Y ) := P Y ) T P Y ). Definition 4. We define recursively a vector τ R 2 and an s.l.. ε with oerations in Ω := { +,,, i, / i} R S R c. Initially, ε 0 := 0 and τ 0 := y 0, 1). Then, we define ε i and τ i recursively on i with 1 i k by: if Γ i = a c ; ), then ε i := 0, τ i := a, 0); if Γ i = a ; u), then ε i := a ε u, τ i := aτ u ; if Γ i = ±; u, v), then ε i := ε u ± ε v, τ i := τ u ± τ v ; if Γ i = ; u, v) and we denote by τ u = a, A), τ v = b, B), then τ i = ab, ab + ba) and ε i equals ε u ε v + A ε v + B ε u ) /) Z y 0)) + a ε v + b ε u ) + 2 AB) Z y 0) /) 2). 1) Recall that multilications denoted by are the ones between -adics. Finally, we set ε P := ε k and τ P := τ k where k is the number of instructions in the s.l.. P. Lemma 14. The s.l.. ε P is a shifted algorithm for E P and y 0. Its multilicative comlexity is bounded by 2L + 1. Also, τ P is the vector of coefficients of the olynomial T P in the basis 1, Y y 0)). Proof. Let us call P i the ith result of the s.l.. P on the inut Y over R [Y ], with 0 i k. We denote by E i := E Pi and T i := T Pi for all 0 i k. Let us rove recursively that ε i is a shifted algorithm for E i and y 0, and that τ i is the vector of coefficients of T i in the basis 1, Y y 0)). For the initial ste i = 0, we have P 0 = Y and we verify that E 0 Y ) = ε 0 Y ) = 0 and T 0 Y ) = y 0 + Y y 0). The s.l.. ε 0 is executable on y over R and its shift is +. Now we rove the result recursively for i > 0. We detail the case when Γ i = ; u, v), the others cases being straightforward. Equation 1) corresonds to the last equation of P i = P up v E i = E u + T u ) E v + T v ) T i E i = E u E v + [T v E u + T u E v ] + T u T v T i) E i = E u E v + [P u y 0) E v + P v y 0) E u ) Y y 0) + P u y 0) E v + P v y 0) E u )] +P u y 0) P v y 0) Y y 0) 2. Also τ i = P u y 0) P v y 0), P u y 0) P v y 0) + P v y 0) P u y 0)). The s.l.. ε i is executable on y over R because, for all j < i, sh ε j) > 0 imlies that Aε v y) + Bε u y)) / R. Concerning the shifts, since sh ε u), sh ε v) > 0, we can check that every oerand in equation 1) has a ositive shift. So sh ε i) > 0. Then, take i = r to conclude the roof. Concerning multilicative comlexity, we slightly change ε 0 such that it comutes once and for all Y y 0) /) 2 before returning zero. Then, for all multilication instructions in the s.l.. P, the s.l.. ε P adds two multilications between -adics see equation 1)). So L ε P ) = 2L + 1. Proosition 15. Let P be a univariate olynomial over R given as an s.l.. whose multilicative comlexity is L. Then, the following algorithm Ψ : Z P y0) + P y 0) y 0 ε P Z) P y 0) is a shifted algorithm associated to Φ and y 0 whose multilicative comlexity is 2L + 1. Proof. We have Φ Y ) = Ψ Y ) over the algebra K [Y ] because Φ Y ) = P y 0) + P y 0) y 0 + E P Y )) /P y 0). Because of Lemma 14 and ν P y 0)) = 0, the s.l.. Ψ is executable on y over R and its shift is ositive. We conclude with L Ψ) = L ε P ) = 2L + 1 as the division by P y 0) is an oeration in the set S. Remark 16. By adding the square oeration 2 to the set of oerations Ω of P, we can save a few multilications. In Definition 4, if Γ i = 2 ; u ) and τ u = a, A), then we define ε i by ε u ε u + 2 a + A Z y 0)))+ 2 A 2 Z y 0) /) 2). Thereby, we reduce the multilicative comlexity of ε P and Ψ by the number of square oerations in P.

6 Theorem 17. Let P R[Y ] and y 0 R/ ) be such that P y 0) = 0 mod and P y 0) 0 mod. Denote by y R the unique solution of P lifted from y 0. Assume that P is given as an s.l.. with oerations in Ω := {+,, } R R c whose multilicative comlexity is L. Then, we can lift y u to recision n in time 2L + 1) R n) + On). Proof. By Proositions 11 and 15, y can be comuted as a recursive -adic number with the shifted algorithm Ψ. Proosition 10 gives the announced comlexity. Remark 18. We can imrove the bound on the multilicative comlexity when the olynomial has a significant art with ositive valuation. Indeed suose that the olynomial P is given as P Y ) = α Y ) + β Y ) with α and β two s.l.. s. Then the art β Y ) is already shifted. In this case, set ε P := ε α + β so that Ψ : Z α y0) + α y 0) y 0 ε P Z) α y 0) is a shifted algorithm for P with multilicative comlexity 2L α) + L β) LINEAR ALGEBRA OVER P-ADICS As an extension of the results of the revious section, we will lift a simle root of a system of r algebraic equations with r unknowns in Section 5. For this matter, one needs to solve a linear system based on the Jacobian matrix in a relaxed way, as we describe in this section. Any matrix A M r s R ) can be seen as a -adic matrix, i.e. a -adic number whose coefficients are matrices over M. In this case, the matrix of order n will be denoted by A n M r s M), so that A = n=0 Ann. 4.1 Inversion of a scalar matrix We can generalize the remark of [1, Section 6.1]: because of the roagation of the carries, the comutation of the inverse of a regular r r matrix with coefficients in M is not immediate in the -adic case. Let us recall the scalar case. We define two oerators mul rem and mul quo such that for all β M and a R, mul remβ, a) n = remβa n, ) and mul quoβ, a) n = quoβa n, ) M. Thus βa = mul rem β, a) + mul quo β, a). We shall introduce two oerators Mul rem and Mul quo which are the matrix counterarts of both oerators mul rem and mul quo. Let B M r M) and A M r s R ) seen as a -adic matrix, then the nth term of Mul rem B, A) is rem BA n, ) M r s M), while the one of Mul quo B, A) corresonds to quo BA n, ) M r s R ), so that we have BA = Mul rem B, A) + Mul quo B, A). Let us denote by MM r, s) the number of oerations in the ground ring to multily a square matrix of size r and a matrix of size r s. Recall that if r s, then MM r, s) O r 2 s ω 2) and otherwise MM r, s) O r ω 1 s ), where ω is the exonent of matrix multilication over any ring. Lemma 19. Let B and A be two -adic matrices such that B M r M) and A M r s R ). Then, the comutations of Mul rem B, A), Mul quo B, A), and therefore of BA, can be done to recision n in time O MMr, s)n). Proof. To comute Mul rem B, A), we multily B and A as if B were over R/ ) and A over R/ )) [[x]]. To comute Mul quo B, A), for each k < n, we multily B with A k and only kee the quotient by of this roduct. Therefore, the total cost is in O MM r, s) n). Proosition 20. Let A be a relaxed matrix of size r s over R and let B M r M). If B is invertible modulo and Γ:=B 1 mod, then the roduct C = B 1 A satisfies C = Mul rem Γ, A Mul quo B, C)) 2) with C 0 = ΓA 0 mod. Furthermore, C can be comuted u to recision n in time O MM r, s) n + MMr, r)). Proof. First, Γ is comuted in time OMMr, r)). Then A = Mul rem B, C)+Mul quo B, C) so we can deduce that Mul rem B, C) = A Mul quo B, C). It remains to multily both sides by Γ using Mul rem to rove equation 2). From the definition of Mul quo, we can see that the nth term of the right-hand side of equation 2) involves only C n 1. So C is recursively comuted with a cost evaluated in Lemma Inversion of a matrix over -adics We can now aly the division of matrices over -adic integers, as in [12]. Proosition 21. Let A M r s R ) and B M r R ) be two relaxed matrices such that B 0 is invertible of inverse Γ = B 1 0 mod. Then, the roduct C = B 1 A satisfies )) C = B 1 B B0 0 A C, 3) with C 0 = ΓA 0 mod. Thus, C can be comuted u to recision n in time MM r, s) Rn) + On)) + O MMr, r)). Proof. The right-hand side of equation 3) is a shifted algorithm associated to C A BC and Γ. The only -adic matrix roduct involves MMr, s) -adic multilications and therefore a cost of MM r, s) Rn). The cost of the roduct by B 1 0 is estimated in Proosition 20. Remark 22. Notice that if B M r R), then the -adic multilications in the roof of Proosition 21 are in R R, so that C = B 1 A costs This is analogous to the inversion of matrices with olynomial entries which can be done in time linear in the recision [20]. 5. MULTIVARIATE ROOT LIFTING In this section, we lift a -adic root y R r of a olynomial system P = P 1,..., P r) R [Y] r = R [Y 1,..., Y r] r in a relaxed recursive way. We make the assumtion that y 0 = y 1,0,..., y r,0) R/ )) r is a regular modular root of P, i.e. its Jacobian matrix dp y0 is invertible in M r R/ )). The Newton-Hensel oerator ensures both the existence and the uniqueness of y R r such that P y) = 0 and y 0 = y mod. From now on, P is a olynomial system with coefficients in R and y R r is the unique root of P lifted from the modular regular root y 0 R/ )) r. Proosition 23. The olynomial system Φ Y) := dp 1 y 0 dp y0 Y) P Y)) K [Y] r allows the comutation of y.

7 Proof. We adat the roof of Proosition 11. dφ y0 = 0, Φ allows the comutation of y. Since As in the univariate case, we have to introduce a ositive shift in Φ. In the following, we resent how to do so deending on the reresentation of P. 5.1 Dense algebraic systems In this subsection, we assume that the algebraic system P is given in dense reresentation. We assume that d 2, where d := max 1 i,j r deg Xj P i))+1, so that the dense size of P is bounded by rd r. As in the univariate case, the shift of Φ Y) is 0. We adat Lemma 12 and Proosition 13 to the multivariate olynomial case as follows. For 1 j k r, let Q j,k) be olynomial systems such that P Y) equals P y 0)+dP y0 Y)+ Q j,k) Y) Y j y j,0) Y k y k,0 ). 1 j k r Algorithm 2 - Dense olynomial system root lifting Inut: P R [Y] r with a regular root y 0 in R/ )) r. Outut: A shifted algorithm Ψ associated to Φ and y For 1 j k r, comute a Q j,k) Y) from ) P Y) ) Zj y 2. For 1 j k r, let r j,k Z) := j,0 Zk y k,0 3. Let Ψ 1 : Z 1 j k r Qj,k) Z) r j,k Z) 4. return the shifted algorithm Ψ : Z dp 1 y 0 P y0) dp y0 y 0) + 2 Ψ 1 ). Theorem 24. Let P = P 1,..., P r) be a olynomial system in R [Y] r in dense reresentation, satisfying d 2 where d := max 1 i,j r deg Xj P i)) + 1, and let y 0 be an aroximate zero of P. Then Algorithm 2 oututs a shifted algorithm Ψ associated to Φ and y 0. The recomutation in Ψ costs Õ rdr ), while comuting y u to recision n costs rd r R n) + Or 2 n + r ω ). Proof. First, for j r, we erform the Euclidean division of P by Y j y j,0) 2 to reduce the degree in each variable. The naïve algorithm does the first division in time Ord r ). Then the second division costs Or2d r 1 ) because we reduce a olynomial with less monomials. The third Or2 2 d r 2 ) and so on. At the end, all these divisions are done in time Ord r ). Then, for each P i, it remains a olynomial with artial degree at most 1 in each variable. Necessary divisions by Y j y j,0) Y k y k,0 ) are given by the resence of a multile of Y jy k, which gives rise to a cost of O 2 r ) = ord r ). Next, we have to evaluate Ψ 1 at y. Since the total numbers of monomials of Q j,k) Y) for 1 j k r is bounded by rd r, Proosition 10 gives the desired cost estimate for the evaluation of y at recision n. Finally, we have to multily this by the inverse of the Jacobian of P at y 0, which is a matrix with coefficients in R. By Proosition 20 and Remark 22, and since we only lift a single root, it can be done at recision n in time Or 2 n + r ω ). 5.2 Algebraic systems as s.l.. s In this subsection, we assume that the algebraic system P is given as an s.l.. We kee basically the same notations as in Section 3.2. Given an algebraic system P, we define T P Y) := P y 0) + dp y0 Y y 0) and E P Y) := P Y) T P Y). We adat Definition 4 so that we may define τ and ε for multivariate olynomials. Definition 5. We define recursively τ i R R, ε i R for 1 i r with oerations in Ω := { +,,, j, / j} R S R c. Initialize ε r+j i := 0, τ r+j i := y j,0, y j y j,0) for all 1 j r. Then for 1 j k i where k i is the number of instructions in the s.l.. P i, we define ε j i and τ j i recursively on j by formulas similar to Definition 4. Let us detail the changes when Γ j = ; u, v): Let τi u = a, A) and τi v = b, B), then define τ j i by ab, a B + b A) and ε j i by a + A + ε u i ) εv i + b + B) εu i As before, we set ε Pi := ε k i i ) + 2 A B ). and τ Pi := τ k i i. Lemma 25. If τ Pi = a, A) then a = P iy 0) and A = dp i) y0 Y y 0) R. Besides, ε P := ε P1,..., ε Pr ) is a shifted algorithm for E P and y 0 whose comlexity is 3L. Proof. Following the roof of Lemma 14, the first assertion is clear, as is the fact that ε P is a shifted algorithm for E P and y 0. Finally, for all instructions in the s.l.. P i, ε Pi adds three multilications between -adics see oerations in formulas above). So L ε P ) = 3L. Theorem 26. Let P be a olynomial system of r olynomials in r variables over R, given as an s.l.. such that its multilicative comlexity is L. Let y 0 R/ )) r be such that P y 0) = 0 mod and det dp y0 ) 0 mod. Denote by y the unique solution of P lifted from y 0. Then, the algorithm Ψ : Z dp 1 y 0 P y 0) + dp y0 y 0) ε P Z)) is a shifted algorithm associated to Φ and y 0. Thus, one can comute y to recision n in time 3L Rn) + Or 2 n + r ω ). Proof. Similarly to Proosition 15, Ψ is a shifted algorithm. By Proosition 10 and Lemma 25, one can evaluate ε P y) in time 3L Rn) + On). Besides, by Remark 22, the oeration dp 1 y 0 ) costs Or 2 n + r ω ). 6. IMPLEMENTATION AND TIMINGS In this section, we dislay comutation times in milliseconds for the univariate olynomial root lifting and for the comutation of the roduct of the inverse of a matrix with a vector or with another square matrix. Timings are measured using one core of an Intel Xeon X5650 at 2.67 GHz running Linux, Gm [9] and setting = a 30 bit rime number. Our imlementation is available in the files whose names begin with series_carry or _adic in the C++ library algebramix of Mathemagix. In the following tables, the first line, Newton corresonds to the classical Newton iteration [8, Algorithm 9.2] used in the zealous model. The second line Relaxed corresonds to our best variant. The last line gives a few details about which variant is used. We make use of the naive variant N and the relaxed variant R. Furthermore, when the recision is high, we make use of blocks of size 32 or 1024.

8 That means, that at first, we comute the solution f u to recision 32 as F 0 = f f with the variant N. Then, we say that our solution can be seen as a 32 -adic integer F = F F n 32n + and the algorithm runs with F 0 as the initial condition. Then, each F n is decomosed in base to retrieve f 32n,..., f 32n+31. Although it is cometitive, the initialization of F can be quite exensive. BN means that F is comuted with the variant N, while BR means it is with the variant R. Finally, if the recision is high enough, one may want to comute F with blocks of size 32, and therefore f with blocks of size B 2 N res. B 2 R ) means that f and F are comuted u to recision 32 with the variant N and then, the adic solution is comuted with the variant N res. R ). Polynomial root. This table corresonds to the lifting of a regular root from F to Z at recision n as in Section 3.1. Dense olynomial of degree 127 n Newton Relaxed Variant R BN BN BR BR BR BR In this table, the timings of Newton are always better than Relaxed. However, if the unknown required recision is slightly above a ower of 2, e.g. 2 l + 1, then one needs to comute at recision 2 l+1 with Newton algorithms. Whereas relaxed algorithms increase the recision one by one. So the timings of Relaxed are better on significant ranges after owers of 2. Linear algebra. The next two tables corresond to timings for comuting B 1 A at recision n, with A, B M r rz ). In this case, Relaxed erforms well. Square matrices of size r = 8 n Newton Relaxed Variant N N N BN BN BN B 2 N B 2 N Square matrices of size r = 128 n Newton Relaxed Variant N N N BN BN Now, we solve integer linear systems and retrieve the solutions over Q, using the rational number reconstruction [8, Section 5.10]. We set q as to the ower 2 j and ick at random a square matrix B of size r with coefficients in M = {0,..., q 1}. We solve BC = A with a random vector A. Because we deal with q-adic numbers at low recision, we only use the variant N. We comared with Linbox [19] and IML [4] but we do not dislay the timings of IML within Linbox because they are about 10 times slower. As Linbox and IML are designed for big matrices and small integers, it is not surrising that Relaxed erforms better. Integer linear system of size r = 4 j Linbox Relaxed Integer linear system of size r = 32 j Linbox Relaxed In fact, when j 3, there is a major overhead coming from the use of Gm. Indeed, in these cases, we transform q-adic numbers into -adic numbers, comute u to the necessary recision and call the rational reconstruction. Acknowledgments We would like to thank J. van der Hoeven, M. Giusti, G. Lecerf, M. Mezzarobba and É. Schost for their helful comments and remarks. For their hel with Linbox, we thank B. Boyer and J.-G. Dumas. We would like to thank the three anonymous referees for their helful comments. 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