Desingularization Explains Order-Degree Curves for Ore Operators

Transcription

1 Desingularization Exlains Order-Degree Curves for Ore Oerators Shaoshi Chen Det. of Mathematics / NCSU Raleigh, NC 27695, USA schen2@ncsu.edu Maximilian Jaroschek RISC / Joh. Keler University 4040 Linz, Austria mjarosch@risc.jku.at Michael F. Singer Det. of Mathematics / NCSU Raleigh, NC 27695, USA singer@ncsu.edu Manuel Kauers RISC / Joh. Keler University 4040 Linz, Austria mkauers@risc.jku.at ABSTRACT Desingularization is the roblem of finding a left multile of a given Ore oerator in which some factor of the leading coefficient of the original oerator is removed. An order-degree curve for a given Ore oerator is a curve in the r, d)-lane such that for all oints r, d) above this curve, there exists a left multile of order r and degree d of the given oerator. We give a new roof of a desingularization result by Abramov and van Hoeij for the shift case, and show how desingularization imlies order-degree curves which are extremely accurate in examles. Categories and Subject Descritors I..2 [Comuting Methodologies]: Symbolic and Algebraic Maniulation Algorithms General Terms Algorithms Keywords Ore Oerators, Singular Points. INTRODUCTION We consider linear oerators of the form L l 0 + l + + l r r, Suorted by the National Science Foundation NSF) grant CCF Suorted by the Austrian Science Fund FWF) grant Y464-N8. 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 3, June 26 29, 203, Boston, Massachusetts, USA. Coyright 203 ACM /3/06...$0.00. where l 0,..., l r are olynomials or rational functions in x, and denotes, for instance, the derivation d or the shift dx oerator x x +. Formal definitions are given later.) Oerators act in a natural way on functions. They are used in comuter algebra to reresent the functions f which they annihilate, i.e., L f 0. Multilication of oerators is defined in such a way that the roduct of two oerators acts on a function like the two oerators one after the other: P L) f P L f). Therefore, if L is an annihilating oerator for some function f, and if P is any other oerator, then P L is also an annihilating oerator for f. We are interested in turning a given oerator L into a nicer one by multilying it from the left by a suitable P, for two different flavors of nice. First, we consider the roblem of removing factors from the leading coefficient l r of L. This is known as desingularization and it is needed for comuting the values of f at the roots of l r rovided it is defined there). Desingularization of differential oerators is classical [9], and for difference oerators, Abramov and van Hoeij [2, ] give an algorithm for doing it. We give below a new roof of a slightly generalized version of) their results. Secondly, we consider the roblem of roducing left multiles with olynomial coefficients of low degree. Unlike the situation for commutative olynomials, a left multile P L of L may have olynomial coefficients even if P has rational function coefficients with nontrivial denominators and the olynomial coefficients of L have no common factors. In such situations, it may haen that the degrees of the olynomial coefficients in P L are strictly less than those in L. This henomenon can be exloited in the design of fast algorithms because a small increase of the order can allow for a large decrease in degree and therefore yield a smaller total size of the oerator trading order for degree ). Degree estimates suorting this technique have been recently given for a number of different comutational roblems [4, 7, 6, 3]. Although limited to secial situations, these estimates can overshoot by quite a lot. Below we derive a general estimate for the relation between orders and degrees of left multiles of a given oerator L from the results about desingularization. This estimate is indeendent of the context from which the oerator L arose, and it is fairly accurate in examles.

2 2. OVERVIEW Before discussing the general case, let us illustrate the concets of desingularization and trading order for degree on a concrete examle. Consider the differential oerator L x 35x 2 x 3 + 2x 4 ) x 3x 2 x 3 ) + x)23 20x x 2 + 2x 3 ) 2 Q[x][ ], d. dx where That L is desingularizable at a root of) : 23 20x x 2 + 2x 3 means that there is some other oerator P Qx)[ ] such that P L has coefficients in Q[x] and its leading coefficient no longer contains as factor. Such a P is called a desingularizing oerator for L at and P L the corresonding desingularized oerator. In our examle, P x 36x2 + Qx)[ ] is a desingularizing oerator for L at, the desingularized oerator is P L x + 04x x 3 ) x + 36x 2 ) x 240x 2 36x 3 ) x) 3. A desingularizing oerator need not exist. For examle, it is imossible to remove the factor x + from the leading coefficient of L by means of desingularization. In Section 3 we exlain how to check for a given oerator L and a factor of its leading coefficient whether a desingularizing oerator exists, and if so, how to comute it. Desingularization causes a degree dro in the leading coefficient but may affect the other coefficients of the oerator in an arbitrary fashion. However, a desingularizing oerator can be turned into an oerator which lowers the degrees of all the coefficients. To this end, multily P from the left by some olynomial q Q[x] for which the coefficients of qp have low degree modulo, i.e., for which qp P + P2 where P, P 2 Q[x][ ] and P has low degree coefficients. In our examle, a good choice is q x)/299, i.e. P 22x + 29) x), P Since P L has olynomial coefficients, so does PL qp L P2L 0 65x + 22x 2 ) x 34x 2 ) x 22x 2 ) 2 + x)43 34x) 3. This oerator has degree deg x L) + deg x P ) deg x ) 2, comared to deg x L) + deg x P ) 3 achieved with the original desingularizing oerator. There is no left multile of L of order 3 and degree less than 2. There is also none of order 4 and degree less than 2, but there does exist a left multile of degree and order 5. It can be obtained from P by multilying from the left by an oerator q 0 + q + q 2 2 Q[x][ ] of order 2 for which the coefficients of 3 q 0 + q + q 2 2 )P have low degrees modulo 3 : Taking q x 88x2 8940, q 87x2 +3x 92) 8940, q we have q 0 + q + q 2 2 )P 3 Q + Q 2, where Q x 436x 2 48x x 4 + 2x 5 4x 6 ) x 752x x x 4 2x 5 4x 6 ) + x 7) 9 + x + 2x 2 ) , Q x) x) Set Q : 3 Q. Then, since P L has olynomial coefficients, so does QL q 0 + q + q 2 2 )P L Q 2L 2 + x) x) 8 + 2x) x) x) x) 5. Its degree is deg x L) + deg x Q ) 3 deg x ). As the factor x + cannot be removed from L, we cannot hoe to reduce the degree even further. We have thus found that the region of all oints r, d) N 2 such that there is a left Qx)[ ]-multile of L of order r and with olynomial coefficients of degree at most d is given by 2, 4) + N 2 ) 3, 2) + N 2 ) 5, ) + N 2 ). In Section 4 we exlain the construction of the oerators Q that turn a desingularizing oerator into one that lowers all the degrees as far as ossible, and we give a formula that describes the oints r, d) for which such a Q exists. 3. PARTIAL DESINGULARIZATION In this section we discuss under which circumstances an oerator L admits a left multile P L in which a factor of the leading coefficient of L is removed. This is of interest in its own right, and will also serve as the starting oint for the construction described in the following section. In view of this latter alication, we cover here a slightly generalized variant of desingularization, which not only alies to the case where a factor can be comletely removed, but also cases where only the multilicity of the factor can be lowered. Examle. In the shift case i.e., x x+) ), consider the oerator L 3 + x)9 + 7x + x 2 ) x + 47x 2 + 2x 3 + x 4 ) x) x + x 2 ) 2. The factor x + 2) 2 in the leading coefficient cannot be removed comletely. Yet we can find a multile in which x + 2 aears in the leading coefficient in shifted form) with multilicity one only. One such left multile of L is x + 25x 2 ) x + 258x x 3 ) x + 83x x 3 ) x) 3. We seak in this case of a artial desingularization. The general definition is as follows. We formulate it for oerators in an arbitrary Ore algebra O : A[ ] : A[ ; σ, δ] where A is a K-algebra in our case tyically A K[x] or A Kx)), K is a field, σ : A A is an automorhism and δ : A A a σ-derivation, i.e., a K-linear ma satisfying the skew Leibniz rule δq) δ)q + σ)δq) for, q A. For any f A, the multilication rule in A[ ; σ, δ] is f σf) + δf). We write deg L) for the order of L A[ ], and if A K[x], we write deg x L) for the maximum degree among the

3 olynomial coefficients of L. For general information about Ore algebras, see [5]. Definition 2. Let L K[x][ ; σ, δ] and let K[x] be such that lc L) K[x]. We say that is removable from L at order n if there exists some P Kx)[ ] with deg P ) n and some w, v K[x] with gcd, w) such that P L K[x][ ] and σ n lc P L)) w v lc L). We then call P a -removing oerator for L, and P L the corresonding - removed oerator. is simly called removable from L if it is removable at order n for some n N. If gcd, lc L)/v)), we say desingulariz[able ing ed] instead of remov[able ing ed], resectively. The backwards shift σ n in the definition above is introduced in order to comensate the effect of the term n in P on the leading coefficient on L i.e., lc n L) σ n lc L)).) Moreover, observe that in this definition, removing a olynomial does not necessarily mean that the -removed oerator has no roots of some shift of) in its leading coefficient. If L contains some factors of higher multilicity, as in the examle above, then removal of a olynomial is defined so as to resect multilicities. Also observe that in the definition we allow that some new factors w are introduced when is removed. This is only a matter of convenience. We will see below that we may always assume v w, i.e., if something can be removed at the cost of introducing new factors into the leading coefficient, then it can also be removed without introducing new factors. The justification rests on the following lemma. Lemma 3. Let L K[x][ ; σ, δ], let K[x] with lc L) be removable from L, and let P Kx)[ ; σ, δ] be a -removing oerator for L with deg P ) n.. If U K[x][ ] with gcdlc U), σ n+deg U) )), then UP is also a -removing oerator for L. 2. If P P +P 2 for some P Kx)[ ] with deg P ) n and P 2 K[x][ ], then P is also a -removing oerator for L. 3. There exists a -removing oerator P with deg P ) n and with σ n lc P L)) lc L). Proof. Let v, w K[x] be as in Definition 2, i.e., gcd, w) and vσ n lc P L)) w lc L).. Since P L is an oerator with olynomial coefficients, so is UP L. Furthermore, with u lc U) and m deg U) we have vσ n m lc UP L)) σ n m u)w lc L). Since gcdu, σ n+m )), we have gcdσ n m u)w, ), as required. 2. Clearly, P 2 K[x][ ] imlies P 2L K[x][ ]. Since also P L K[x][ ], it follows that P L P P 2)L P L P 2L K[x][ ]. If deg P 2) < n, then we have lc P L) lc P L), so there is nothing else to show. If deg P 2) n, then lc P L) lc P L) lc P 2L) and therefore vσ n lc P L)) vσ n lc P L) lc P 2L)) w vσ n lc P 2))) lc L). Since gcd, w vσ n lc P 2))) gcd, w), the claim follows. 3. By the extended Euclidean algorithm we can find s, t K[x] with sw + tv. Then σ n s)p is -removing of order n by art σ n s) is obviously corime to σ n )), and its leading coefficient is σ n sw v ) σ n v) σn t). By art 2 we may discard the olynomial art σ n t), obtaining a -removing oerator P with deg P ) n and σ n lc P L)) v lc L). Using art, we can obtain from here an oerator with the desired roerty. The lemma imlies that if there is a -removing oerator at all, then there is also one in which all the denominators are owers of σ n ) because any factors corime with can be cleared according to art ), and where all numerators have smaller degree than the corresonding denominators because olynomial arts can be removed according to art 2). Similarly as in the roof of art 3, we can also reduce the roblem of removing a comosite olynomial to the roblem of removing owers of irreducible olynomials. For examle, if 2 is removable from L, where, 2 K[x] are corime, then obviously both and 2 are removable. Conversely, if and 2 are removable, and if P, P 2 are removing oerators of orders n, n 2 with lc P ) /σ n ) and lc P 2) /σ n 2), then for n max{n, n 2} and u, u 2 K[x] with u σ n 2) + u 2σ n ) gcdσ n ), σ n 2)) the oerator P : u n n P + u 2 n n 2 P 2 Kx)[ ] is such that P L K[x][ ] and lc P L) lc L)/σ n ). In summary, in order to determine whether a olynomial k k 2 2 km m is removable from an oerator L, it suffices to be able to check for an irreducible olynomial i and a given k i whether k i i is removable. Let now be an irreducible olynomial and k. If there exists a k -removing oerator, then it can be assumed to be of the form 0 P σ n ) n e 0 σ n ) e σ n ) e n + n σ n ) k n, for some e 0,..., e n N, and 0,..., n K[x] with deg x i) < e i deg x ). In order to decide whether such an oerator exists, it is now enough to know a bound on n as well as a bound e on the exonents e i, for if n and e are known, we can make an ansatz i e j0 i,jxj with undetermined coefficients i,j, then calculate P L and rewrite all its coefficients in the form a/σ n ) e + b for some olynomials a, b deending linearly on the undetermined i,j, then comare the coefficients of the various a s with resect to x to zero and solve the resulting linearly system for the i,j. How the bounds on n and e are derived deends on the articular Ore algebra at hand. In this aer, we give a comlete treatment of the shift case σ)x) x + ), δ 0) and make some remarks about the differential case σ id, δ d ). For other cases, see the rerint [8]. dx 3. Shift Case In this section, let K[x][ ] denote the Ore algebra of recurrence oerators, i.e., σ is the automorhism maing x

4 to x + and δ is the zero ma. This case was studied by Abramov and van Hoeij [2, ]. We give below a new roof of their result, and extend it to the case of artial desingularization. For consistency with the differential case, we formulate the result for the leading coefficients, while Abramov and van Hoeij consider the analogous for the trailing coefficients. Of course, this difference is immaterial. We roceed in two stes. First we give a bound on the order of a removing oerator Lemma 4), and then, in a second ste, we rovide a bound on the exonents in the denominators Theorem 5). As exlained above, it is sufficient to consider the case of removing owers of irreducible olynomials, and we restrict to this case. Lemma 4. In the oerator algebra K[x][ ] with the commutation rule x x + ), let L l 0 + l + + l r r K[x][ ] with l 0, l r 0, and let be an irreducible factor of lc L) such that k is removable from L for some k. Let n N be s.t. gcdσ n ), l 0) and gcdσ m ), l 0) for all m > n. Then k is removable at order n from L. Proof. By assumtion on L, there exists a k -removing oerator P, say of order m, and by the observations following Lemma 3 we may assume that 0 P σ m ) + e 0 σ m ) + + m e σ m ) em m, for e i N and i K[x] with deg x i) < e i deg x ) i 0,..., m). We may further assume gcdσ m ), i) for i 0,..., m viz. that the e i are chosen minimally). Suose that m > n. We show by induction that then e 0 e e m n 0, so that i 0 for i 0,..., m n, i.e., the oerator P has in fact the form m n m P σ m ) e m n + + m n σ m ) em m. Thus n m P Kx)[ ] is a k -removing oerator of order n. Consider the oerator T : r+m i0 ti i : P L K[x][ ]. From 0 σ m ) e 0 l0 t0 K[x] it follows that e0 0, because gcdσ m ), 0) gcdσ m ), l 0) by the choice of 0 and the assumtion in the lemma, resectively, and this leaves no ossibility for cancellation. Assume now, as induction hyothesis, that e 0 e e i 0 for some i < m n. Then from i t i σ m ) e σi l i 0) + i σ m ) e σi l i 0) i σ m ) e i σi l ) σ m ) e 0 li it follows that σ m ) e i iσ i l 0). By the choice of i we have gcdσ m ), i) and by the assumtion in the lemma we have gcdσ m i ), l 0) because m i > n), so it follows that e i 0. Inductively, we obtain e 0 e e m n 0, which comletes the roof. It can be shown that cannot be removed from L if σ n ) is corime with the trailing coefficient of L for all n N by a variant of [2, Lemma 3.], so the above lemma covers all situations where removing of a factor is ossible. In order to formulate the result about the ossible exonents in the denominator, it is convenient to first introduce some notation. Let us call two irreducible olynomials, q K[x] \ {0} equivalent if there exists n Z such that σ n )/q K. We write [q] for the equivalence class of q K[x] \ {0}. If, q are equivalent in this sense, we write q if σ n )/q K for some n 0, and > q otherwise. The irreducible factors of a olynomial u K[x] can be groued into equivalence classes, for examle u x 4)x ) 3 xx + ) 2 }{{} 2x 5)2x + 3)2 2x + 9) }{{} x 2 + 5x + )x 2 + x + 25) 3 }{{}. For any monic irreducible factor of u K[x], let v u) denote the multilicity of in u, and define v <u) : max{ v qu) q [] : > q }. For examle, for the articular u above we have v x 4u), v <x 4u) 0, v <x+u) 3, and so on. Besides being alicable not only to desingularization but also removal of any factors, the following theorem also refines the corresonding result of Abramov and van Hoeij in so far as their version only covers the case of desingularizing L at some with v <lc L)) 0 whereas we do not need this assumtion. Theorem 5. In the oerator algebra K[x][ ] with the commutation rule x x + ), let L l 0 + l + + l r r K[x][ ] with l 0, l r 0, and let be an irreducible factor of l r such that k is removable from L for some k. Let n N be such that gcdσ n ), l 0) and gcdσ m ), l 0) for all m > n. Then there exists a k -removing oerator P for L and of the form 0 P σ n ) + e 0 σ n ) + + n e σ n ) en n, for some e i N and i K[x] with. deg x i) < e i deg x ) and gcdσ n ), i), and 2. e i k + n v <lc L)) for i 0,..., n, and n, e n k. Proof. Lemmas 3 and 4 imly the existence of an oerator P with all the required roerties excet ossibly the exonent estimate in item 2. Let P be such an oerator, and consider the oerator T : r+n i0 ti i : P L K[x][ ]. Let e max{e,..., e n} and P : n i0 i i : σ n ) e P. Then i σ n ) e e i i i 0,..., n) and σ n ) e T P L and gcd 0,..., n, σ n )). Abbreviating v : v <lc L)), assume that e > k + n v. We will show by induction that then i contains σ n ) with multilicity more than i v for i n, n,..., 0, which is inconsistent with gcd 0,..., n, σ n )). First it is clear that n σ n ) e nσ n l r) contains σ n ) with multilicity e k > nv, because P is k -removing. Suose now as induction hyothesis that there is an i 0 such that σ n ) j v+ j for j n, n,..., i +. Consider the equality σ n ) e t i+r iσ i l r) + i+σ i+ l r ) + + nl r n, where we use the convention l j : 0 for j < 0. The induction hyothesis imlies that σ n ) i+)v+ j for j n, n,..., i +. Furthermore, since i + )v nv < e, we have σ n ) i+)v+ σ n ) e t i+r. Both facts together imly σ n ) i+)v+ iσ i l r). The definition of v ensures that

5 σ n ) is contained in σ i l r) with multilicity at most v, so it must be contained in i with multilicity more than i + )v v i v, as claimed. A referee comments that one can imrove the denominator bounds of terms after the leading term of Theorem 5 as in the code given in the aer [2]. However, this refined bound would not affect our main result, Theorem 9 below. 3.2 Differential Case In this section K[x][ ] refers to the Ore algebra of differential oerators, i.e., σ id and δ d. Let L K[x][ ] and dx suose for simlicity that x is a factor of lc L). In [], the authors show that L can be desingularized at x if and only if x 0 is an aarent singularity, that is, if and only if Ly) 0 admits deg L) linearly indeendent formal ower series solutions. The authors furthermore give an algorithm to find an oerator P such that if ξ is either an ordinary oint of L or an aarent singularity of L, then ξ is an ordinary oint of P L. Therefore this algorithm desingularizes all the oints that can be desingularized. The authors also give a shar bound for deg P ). The authors furthermore give some indications concerning artial desingularizations. Additional information concerning desingularization and the degrees of initial terms can also be found in [0]. 4. ORDER-DEGREE CURVES We now turn to the construction of left multiles of L with olynomial coefficients of small degree, and to the question of how small these degrees can be made. As already indicated in Section 2, we start from an oerator P which removes some factor from the leading coefficient of L, say it removes a olynomial of degree k. According to Lemma 3, we may assume that lc P ) /σ deg P ) ) and that all other coefficients of P are rational functions whose numerators have lower degree than the corresonding denominators. Thus we already have deg x P L) deg x L). Furthermore, if q is any olynomial with deg x q) < deg x ) k, then multilying P by q from left) and removing olynomial arts by Lemma 3 art 2) gives another oerator Q with deg x QL) deg x L). All the oerators Q obtained in this way form a K-vector sace of dimension k. Within this vector sace we search for elements where deg x QL) is as small as ossible. Forcing the coefficients of the highest degrees to zero gives a certain number of linear constraints which can be balanced with the number of degrees of freedom offered by the coefficients of q, as illustrated in the figure below. As long as we force fewer than k terms to zero, we will find a nontrivial solution. degree order If we want to eliminate k terms or more in order to reduce the degree even further, we need more variables. We can create k more variables if instead of an ansatz qp we make an ansatz q 0 + q )P for some q 0, q K[x] with deg x q 0), deg x q ) < k. Again removing all olynomial arts from the rational function coefficients we obtain a vector sace of oerators Q with deg x QL) deg x L) whose dimension is 2k. The additional degrees of freedom can be used to eliminate more high degree terms, the result being an oerator of lower degree but higher order. If we let the order increase further and for each fixed order use all the available degrees of freedom to reduce the degrees to minimize the degrees of the olynomial coefficients, a hyerbolic relationshi between the order and the degree of QL emerges. In Theorem 9 below, we make this relationshi recise, taking into account that for a given oerator L the leading coefficient may contain several factors that are removable at different orders n. The resulting region of all oints r, d) N 2 for which there exists a left multile of L of order r with olynomial coefficients of degree at most d is then given by an overlay of a finite number of hyerbolas. Before turning to the roof of this theorem, let us illustrate its basic idea with the examle oerators from Section 2. Examle 6. Let L Q[x][ ], Q[x], and P Qx)[ ] be as in Section 2. Recall that is an irreducible cubic factor of lc L) and that P is a -removing oerator for L. We have P + 0 for some, 0 Q[x] with deg x ) 0 and deg x 0) 2. We have seen in Section 2 that there is an oerator Q Qx)[ ] of order 3 such that QL Q[x][ ] and deg x QL). Our goal here is to exlain why this oerator exists. Make an ansatz Q q 0 + q + q 2 2 )P with undetermined olynomials q 0, q, q 2. After exanding the roduct and alying commutation rules, Q has the form q )q2 +...)q )q2 +...)q +...)q )q2 +...)q +...)q0 3, where the...) are certain olynomials whose recise form is irrelevant for our urose. Note that by Lemma 3 art ), Q L Q[x][ ] regardless of the choice of q 0, q, q 2, and that by Lemma 3 art 2), this roerty is not lost if we add to Q some oerator in Q[x][ ] of our choice. Therefore, if Q 2 Q[x][ ] is the oerator obtained from 3 Q Q[x][ ] by reducing all the coefficients modulo 3, then 3 Q 2L Q[x][ ], still regardless of the choice of q 0, q, q 2. The coefficients of Q 2 deend linearly on the undetermined olynomials q 0, q, q 2. If we choose their degree to be deg x ) 2, then we have 32 + ) 9 variables for the coefficients of q 0, q, q 2. Choosing a higher degree would give more variables but also introduce undesired solutions such as q 0 q q 2, for which the reduction modulo 3 leads to the useless result Q 2 0. This cannot haen if we enforce deg x q i) < deg x ). The oerator 3 Q 2L has degree deg x Q 2) + deg x L) 3 deg x ) deg x Q 2) 5, which is equal to if deg x Q 2) 6. A riori, the degree of Q 2 in x may be u to deg x 3 ) 8. In order to bring it down to 6, we equate the coefficients of x i j for i 7, 8 and j 0,..., 3 to zero. This gives 8 equations. As there

6 are more variables than equations, there must be a nontrivial solution. For formulating the roof of the general statement, it is convenient to work with an alternative formulation of removability, which is rovided in the following lemma. Throughout the section, K[x][ ] K[x][ ; σ, δ] is an arbitrary Ore algebra. Lemma 7. K[x] is removable from L K[x][ ] at order n if and only if there exists P K[x][ ] with deg P ) n and P L σ n ) lc P )K[x][ ]. Proof. : P 0 σ n ) lc P is a -removing oerator. P ) : Start from a -removing oerator of the form P 0 n i0 i σ n ) e i i + σ n ) n, and set P σ n ) e P 0 where e max{e 0,..., e n, }. Because of P 0L K[x][ ] it follows that P L σ n ) e K[x][ ] σ n ) lc P )K[x][ ]. The next lemma is a generalization of Bezout s relation to more than two corime olynomials, which we will also need in the roof. Lemma 8. Let u,..., u m K[x] be airwise corime and u u u 2 u m, and let v,..., v m K[x] be such that deg x v i) < deg x u i) i,..., m). If then v v 2 v m 0. v i u u i 0 Proof. Since the u i are airwise corime, u i u/u i for all i. However, u i u/u j for all j i. Both facts together with m viu/ui 0 imly that ui vi for all i. Since deg x vi) < deg x u i), the claim follows. Theorem 9. Let L K[x][ ], and let,..., m K[x] be factors of lc L) which are removable at orders n,..., n m, resectively, so that the σ n i i) are airwise corime. Let r deg L) and m d deg x L) n ) + i degx i), r deg L) + where we use the notation x) + : max{x, 0}. Then there exists an oerator Q Kx)[ ] \ {0} such that QL K[x][ ] and deg QL) r and deg x QL) d. Proof. Let r deg L), and set s : r deg L) so that s deg Q). We may assume without loss of generality n that s is such that i n i > 0 for all i r deg L)+ s+ by simly removing all the i for which n i 0 from s+ consideration. We thus have s n i for all i. Lemma 7 yields oerators P i K[x][ ] of order n i with P il σ n i i) lcp i)k[x][ ]. Set q m s n i j0 σ j+n i i)σ j l i), where l i lc P i). Consider the ansatz Q s n i j0 q i,j q σ j+n i i)σ j l i) j P i for undetermined olynomial coefficients q i,j i,..., m; j 0,..., s n i) of degree less than deg x i). Regardless of the choice of these coefficients, we will always have Q K[x][ ] and Q L qk[x][ ]. Also, for arbitrary R K[x][ ] and Q 2 Q qr we have Q 2 K[x][ ] and Q 2L qk[x][ ]. This means that we can relace the coefficients in Q by their remainders uon division by q without violating any of the mentioned roerties of Q. Also observe that any oerator Q 2 obtained in this way is nonzero unless all the q i,j are zero, because if k is maximal such that at least one of the q i,k is nonzero, then lc Q ) q i,k q σ k+n i i)σ k l i) σk l i) q i,k q σ k+n i i) is nonzero by Lemma 8. Furthermore, lc Q ) 0 mod q because deg x q i,k ) < deg x i) imlies deg x lc Q )) < deg x q). The ansatz for the q i,j gives m s ni+) deg x i) variables. Plug this ansatz into Q and reduce all the olynomial coefficients modulo q to obtain an oerator Q 2 of degree less than deg x q) m s ni + )deg x i) + deg x li)). Then for each of the s+ olynomial coefficients in Q 2 equate the coefficients of the terms x j for ) j > s n i) deg x i) + s n i + ) deg x l i) + ni deg x i) to zero. This gives altogether m ) s n i + ) deg x i) + deg x l ) i) ) s n i) deg x i) + s n i + ) deg x l i) ni deg x i) ) m ) deg x i) ni deg ) x i) equations. The resulting linear system has a nontrivial solution because #vars #eqns s n i + ) deg x i) m ) deg x i) ni deg x i) n i deg x i) ) ni deg x i) > n i deg x i) + n i deg x i) 0. ) )

7 By construction, the solution gives rise to an oerator Q 2 K[x][ ] of order at most n with olynomial coefficients of degree at most s ni) deg x i) + s n i + ) deg x l ) i) + ni deg x i), for which Q 2L qk[x][ ]. Thus if we set Q q Q2 Kx)[ ], we have deg QL) deg L)+s r and deg x QL) is at most deg x L) + deg x Q 2) deg x q) deg x L) + s ni) deg x i) + s n i + ) deg x l ) i) + ni deg x i) s n i + )deg x i) + deg x l i)) deg x L) deg x i) + ni deg x i) m ) deg x L) ni deg x i), as required. The final ste uses the facts x x and x + n x + n for x R and n Z.) Examle 0.. Consider again the examle from Section 2. There we started from an oerator L K[x][ ] of order 2 and degree 4 for which there exists a desingularizing oerator P of order which removes a olynomial of degree 3. According to the theorem, for every r 2 exists an oerator Q K[x][ ] with QL K[x][ ], deg QL) r and ) +3 r + 2 d : deg x QL) 4 r 2 + r. This hyerbola recisely redicts the order-degree airs we found in Section 2: r d Consider the minimal order telescoer L for the hyergeometric term in Examle 6. of [6], L 9x + )3x + )3x + 2) 2 3x + 4)x + ) +... degree 6... ) +... degree 5... ) 2 0x8 + 5x)9 + 5x) + 5x)2 + 5x)x) 3, where reresents the shift oerator and is a certain irreducible olynomial of degree 0. This olynomial is removable of order. Therefore, by the theorem, we exect left multiles of L of order r and degree bounded by ) +0 6r 2 6 r 3 + r 2. In the figure below, the curve d 6 solid) is contrasted with the estimate d 8r dashed) derived last year for this examle as well as the region of all oints r, d) for which a left multile of L of order r and degree d exists gray). The new curve matches recisely the boundary of the gray region, even including the very last degree dro which is not clearly visible on the figure): for r 2 we have 6 7 and for r 3 we have < Consider the minimal order telescoer L for the hyerexonential term in Examle 5.2 in [7]. It has order 3 and degree 40. The leading coefficient contains an irreducible olynomial of degree 23 at order and otherwise only non-removable factors. Theorem 9 therefore redicts left multiles of L of order r and degree 40 r 3 + ) +23 7r r 2 for all r N. Again, this estimate is accurate, while the estimate 24r 9 derived in [7] overshoots Oerators coming from alications tend to have leading coefficients that contain a single irreducible olynomial of large degree which can be removed at order, besides factors that are not removable. But Theorem 9 also covers the more general situation of factors that are only removable of higher order, and even the case of several olynomials that are removable at several orders. As an examle for this general situation, consider the oerator L 8+x)+2x) z) x+26x 2 +7x 3 ) 7 9+3x) 9 2+3x) 2 +x+5x 2 +7x 3 ) 7, where reresents the shift oerator. From its leading coefficient, the olynomial + x + 5x 2 + 7x 3 ) 7 is removable at order, and in addition, + 3x) 7 is removable at order 2. The remaining factors are not removable. According to Theorem 9 we exect that L

8 admits left multiles of order r and degree 32 2 ) ) +, r for all r N. It turns out that this rediction is again accurate for every r. Observe that in this examle the curve is a suerosition of two hyerbolas Although the bound of Theorem 9 aears to be tight in many cases, it is not always tight. The oerator x )x + ) 2 x + 3) 2 x + 5) 2 2x ) + x 2 x + 2) 2 x + 4) 2 x + 6)2x 3), in which denotes the shift, is an examle: It has a left multile of order 2 and degree 3 although according to Theorem 9 we would exect a multile of order 2 to have degree 8 2 ) Thanks to Mark van Hoeij for sharing this examle.) 5. CONCLUSION We believe that removable factors rovide a universal exlanation for all the order-degree curves that have been observed in recent years for various different contexts. We have derived a formula for the boundary of the gray region associated to a fixed oerator L, which, although formally only a bound, haens to be exact in examles coming from alications. This does not immediately imly better comlexity estimates or faster variants of algorithms exloiting the henomenon of order-degree curves, because usually L is not known in advance but rather the desired outut of a calculation, and therefore we usually have no information about the removable factors of lc L). However, we now know what we have to look at: in order to imrove algorithms based on trading order for degree, we need to develo a theory which rovides a riori information about the removable factors of lc L). In other words, our result reduces the task of better understanding order-degree curves to the task of better understanding what causes the aearance of removable factors in oerators coming from alications. 6. ACKNOWLEDGEMENT We thank the referees for their valuable remarks. 7. REFERENCES [] Sergei A. Abramov, Moulay A. Barkatou, and Mark van Hoeij. Aarent singularities of linear difference equations with olynomial coefficients. AAECC, 7:7 33, [2] Sergei A. Abramov and Mark van Hoeij. Desingularization of linear difference oerators with olynomial coefficients. In Proceedings of ISSAC 99, ages , 999. [3] Alin Bostan, Frederic Chyzak, Ziming Li, and Bruno Salvy. Fast comutation of common left multiles of linear ordinary differential oerators. In Proceedings of ISSAC 2, ages 99 06, 202. [4] Alin Bostan, Frédéric Chyzak, Bruno Salvy, Grégoire Lecerf, and Éric Schost. Differential equations for algebraic functions. In Proceedings of ISSAC 07, ages 25 32, [5] Manuel Bronstein and Marko Petkovšek. An introduction to seudo-linear algebra. Theoretical Comuter Science, 57):3 33, 996. [6] Shaoshi Chen and Manuel Kauers. Order-degree curves for hyergeometric creative telescoing. In Proceedings of ISSAC 2, ages 22 29, 202. [7] Shaoshi Chen and Manuel Kauers. Trading order for degree in creative telescoing. Journal of Symbolic Comutation, 478): , 202. [8] Frederic Chyzak, Philie Dumas, Ha Le, Jose Martin, Marni Mishna, and Bruno Salvy. Taming aarent singularities via Ore closure. in rearation. [9] E. L. Ince. Ordinary Differential Equations. Dover, 926. [0] Harrison Tsai. Weyl closure of a linear differential oerator. Journal of Symbolic Comutation, 294 5): , 2000.