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[ x htannihilates that r ][ L f y d ] 3 = P of priori A r 3 of ecom- we , eetmt h re n degree and order the estimate we ) 4 = f M L oee,we h aki to is task the when However, . n degree and y f d n all and -degree h eiaino h bound the of derivation The . for- of For . ( such ,d r, nweg fteehyper- these of knowledge ) † 100 200 300 400 7 d 4 ,w edt predict to need we ], L , d 4 = 3 , · · · · · · · · · · · · · · · · · 8 · · · · · · · · · · · 50 .Hwvr it However, ]. n random a and · · · · · · · · 100 · 2 · from ] sfor ds · r ring hy- ing ra- 4 a ) t justification of the desingularization cost. In practice, it is AP + BlrL = lk for polynomials A, B ∈ C(x)[y] of de- almost always equal to one, and although this is the value to grees at most dL −1 and rP −1, respectively, and com- be expected for generic input, it is surprisingly cumbersome pare coefficients with respect to y. This gives k inho- to give a rigorous proof for this expectation. mogeneous linear systems over C(x) with rP +dL vari- Throughout the paper, we use the following conventions: ables and equations, which only differ in the inhomoge-
neous part but have the same matrix M = Syly(P,lrL ) • C is a field of characteristic zero, C[x] is the usual com- for every k. The claim follows using Cramer’s rule, mutative ring of univariate polynomials over C. We taking into account that the coefficient matrix of the write C[x][y] or C[x,y] for the commutative ring of system has dL many columns with polynomials of de- bivariate polynomials and C[x][∂] for the non-commu- gree dP and rP many columns with polynomials of tative ring of linear differential operators with polyno- degree degx lk(y) = 0 (which is also the degree of the mial coefficients. In this latter ring, the multiplication inhomogeneous part). Note that v = det(M) does not is governed by the commutation rule ∂x = x∂ + 1. depend on k. • L ∈ C[x][∂] is an operator of order rL := deg∂ (L) ′ with polynomial coefficients of degree at most dL := 3. (Multiplication by g ) For each polynomial Q ∈ C[x,y] degx(L). with degy(Q) ≤ rP − 1 there exist polynomials qj ∈ • P ∈ C[x,y] is a polynomial of degrees rP := degy(P ) C[x] of degree at most degx(Q) + 2rP dP such that and dP := deg (P ). It is assumed that P is square-free x rP −1 as an element of C(x)[y] and that it has no divisors in ′ 1 j g Q(x,g)= qj g , ¯ ¯ w lcy(P ) C[y], where C is the algebraic closure of C. j=0 • M ∈ C[x][∂] is an operator such that for every solution X f of L and every solution g of P , the composition f◦g is where w ∈ C[x] is the discriminant of P . To see a solution of M. The expression f◦g can be understood this, first apply Observation 1 (Reduction by P ) to 1 2rP −2 j rewrite −QPx as T = tj y for some either as a composition of analytic functions in the case lcy (P ) j=0 C = C, or in the following sense. We define M such t ∈ C[x] of degree deg (Q)+ d . Then consider an j x PP that for every α ∈ C, for every solution g ∈ C[[x − α]] ansatz AP + BPy = lcy(P )T with unknown polyno- of P and every solution f ∈ C[[x − g(α)]] of L, M mials A, B ∈ C(x)[y] of degrees at most rP − 2 and annihilates f ◦ g, which is a well-defined element of rP − 1, respectively, and compare coefficients with re- C[[x − α]]. In the case C = C these two definitions spect to y. This gives an inhomogeneous linear system coincide. over C(x) with 2rP − 1 variables and equations. The claim then follows using Cramer’s rule.
2. ORDER-DEGREE-CURVE rP Lemma 2. Let u = vw lcy(P ) , where v and w are as in BY LINEAR ALGEBRA the Observations 2 and 3 above. Let f be a solution of L Let g be a solution of P , i.e., suppose that P (x,g(x)) = 0, and g be a solution of P . Then for every ℓ ∈ N there are and let f be a solution of L, i.e., suppose that L(f) = 0. polynomials ei,j ∈ C[x] of degree at most ℓ deg(u) such that Expressions involving g and f can be manipulated according rP −1 rL−1 to the following three well-known observation: ℓ 1 i (j) ∂ (f ◦ g)= ei,j g · (f ◦ g). uℓ i=0 j=0 1. (Reduction by P ) For each polynomial Q ∈ C[x,y] X X ˜ with degy(Q) ≥ rP there exists a polynomial Q ∈ Proof. This is evidently true for ℓ = 0. Suppose it is true ˜ ˜ C[x,y] with degy(Q) ≤ degy(Q) − 1 and degx(Q) ≤ for some ℓ. Then deg (Q)+ dP such that x rP −1 rL−1 ′ ei,j 1 ∂ℓ+1(f ◦ g)= gi · (f (j) ◦ g) Q(x,g)= Q˜(x,g). uℓ i=0 j=0 lcy(P ) X X rP −1 rL−1 ′ ′ ˜ ei,j u − ℓei,j u The polynomial Q is the result of the first step of com- = gi · (f (j) ◦ g) uℓ+1 puting the pseudoremainder of Q by P w.r.t. y. i=0 j=0 X X 2. (Reduction by L) There exist polynomials v, qj,k ∈ ei,j i−1 (j) i (j+1) ′ + ℓ ig · (f ◦ g)+ g · (f ◦ g) g . C[x] of degree at most dLdP such that u rP −1 rL−1 The first term in the summand expression already matches (rL) 1 j (k) f ◦ g = qj,kg · (f ◦ g). the claimed bound. To complete the proof, we show that v j=0 k=0 X X rP −1 rL k i−1 (j) i (j+1) ′ 1 k To see this, write L = l ∂ for some polynomials ig · (f ◦ g)+ g · (f ◦ g) g = qkg (1) k=0 k u lk ∈ C[x] of degree at most dL. Then we have k=0 P X rL−1 for some polynomials qk of degree at most deg(u). Indeed, (rL) −1 (k) (rL) f ◦ g = (lk ◦ g) · (f ◦ g). the only critical term is f ◦ g. According to Observa- lr ◦ g (r ) 1 rP −1 rL−1 j L k=0 tion 2, f L ◦ g can be rewritten as q g · X v j=0 k=0 j,k (k) (f ◦ g) for some qj,k ∈ C[x] of degree at most dLdP . By the assumptions on P , the denominator lrL ◦g can- P P not be zero. In other words, gcd(P (x,y),lrL (y))=1in This turns the left hand side of (1) into an expression of 1 2rP −2 j (k) C(x)[y]. For each k = 0,...,rL −1, consider an ansatz the form v j=0 q˜j,kg · (f ◦ g) for some polynomials P q˜j,k ∈ C[x] of degree at most dLdP . An(rP − 1)-fold appli- 0 ≤ j r(3rP + dL − 1)dP rLrP r ≥ rLrP and d ≥ . r + 1 − rLrP Proof. This is true for ℓ = 0. Suppose it is true for some ℓ. Then Then there exists an operator M ∈ C[x][∂] of order ≤ r and ′ degree ≤ d such that for every solution g of P and every rL−1 ℓ+1 1 (j) solution f of L the composition f ◦ g is a solution of M. In ∂ (f ◦ g)= Eℓ,j (x,g)(f ◦ g) U(x,g)ℓ particular, there is an operator M of order r = rLrP and j=0 ! 2 2 2 2 X degree (3rP + dL − 1)dP rLrP =O (rP + dL)dP rLrP . rL−1 ′ ℓ(Ux + g Uy) (j) = Ei,j · (f ◦ g) Proof. Let g be a solution of P and f be a solution of L. U ℓ+1 j=0 Then we have P (x,g(x)) = 0 and L(f) = 0, and we seek X d r i j 1 ′ (j) 1 ′ (j+1) an operator M = c x ∂ ∈ C[x][∂] such that + ((Eℓ,j )x+g ·(Eℓ,j )y)(f ◦g)+ Eℓ,j g ·(f ◦g) i=0 j=0 i,j U ℓ U ℓ M(f ◦ g)=0. Let r ≥ rLrP and consider an ansatz ′ P P ℓ(Ux+g Uy ) d r We consider the summands separately. In Uℓ+1 , Ux i j M = ci,j x ∂ is already a polynomial in x and g of bidegree at most (2dp − ′ −Px(x,g) 1, 2rP + dL − 1). Since g = and Uy is divisible by i=0 j=0 Py (x,g) X X ′ Py, g Uy is also a polynomial with the same bound for the with undetermined coefficients ci,j ∈ C. Let u be as in Lemma 2. Then applying M to f ◦ g and bidegree. multiplying by ur gives an expression of the form Futhermore, we can write d+r deg(u) r −1 r −1 ′ 1 P L (Eℓ,j )x + g · (Eℓ,j )y = (U(Eℓ,j )x − PxPylr (g)(Eℓ,j)y), i j (k) L qi,j,kx g · (f ◦ g), U i=0 j=0 k X X X=0 where the expression in the parenthesis satisfies the stated bound. where the qi,j,k are C-linear combinations of the undeter- For j + 1 and let g1,...,grP be distinct solutions of P . By r we denote 3. A DEGREEBOUND FOR the C-dimension of the C-linear space V spanned by fi ◦ gj THE MINIMAL OPERATOR for all 1 ≤ i ≤ rL and 1 ≤ j ≤ rP . The order of the operator annihilating V is at least r. We will construct an operator According to Theorem 3, there is operator M of order r = 2 2 of order r annihilating V using Wronskian-type matrices. rLrP and degree d = O((rP + dL)dP rLrP ). Usually there is no operator of order less than rLrP , but if such an operator Lemma 5. There exists a matrix A(x,y) ∈ C[x,y](r+1)×rL accidentally exists, Theorem 3 makes no statement about its such that the bidegree of every entry of the i-th row of A(x,y) degree. The result of the present section (Theorem 8 below) does not exceed (2rdP − i + 1,r(2rP + dL − 1)) and f ∈ V if is a degree bound for the minimal order operator, which also and only if the vector (f,...,f (r))T lies in the column space applies when its order is less than rLrP , and which is better of the (r + 1) × rLrP matrix A(x,g1) · · · A(x,gr ) . than the bound of Theorem 3 if the minimal order operator P has order rLrP . Proof. With the notation of Lemma 4, let A(x,y) be the The following Lemma is a variant of Lemma 2 in which g is r+1−i allowed to appear in the denominator, and with exponents matrix whose (i, j)-th entry is Ei−1,j−1(x,y)U(x,y) . Then A(x,y) meets the stated degree bound. larger than rP − 1. This allows us to keep the x-degrees smaller. By Wi we denote the (r + 1) × rL Wronskian matrix for f1 ◦ gi,...,frL ◦ gi. Then f ∈ V if and only if the vec- Lemma 4. Let f be a solution of L and g be a solution of P . tor (f,...,f (r))T lies in the column space of the matrix N For every ℓ ∈ , there exist polynomials Eℓ,j ∈ C[x,y] for W1 · · · WrP . Hence, it is sufficient to prove that Wi r (r) and A(x,gi) have the same column space. The following Hence f ∈ V ⇔ det C1f ±··· + (−1) det Cr+1f = 0. matrix equality follows from the definition of Ei,j Due to Lemma 7, the latter condition is equivalent to c1f + (r) ℓ−1 rL · · ·+cr+1f = 0, where cℓ =(−1) det Cℓ/ (gi −gj ) . f1 ◦ gi · · · frL ◦ gi i Lemma 7. For every 1 ≤ ℓ ≤ r + 1 the determinant of Cℓ obtain the expression in the following conjecture. rL is divisible by i Aℓ(x,g1)−Aℓ(x,g2) Aℓ(x,g2) · · · Aℓ(x,grP ) 2 2 det Cℓ = . − rLrP − rLdLdP , B(g1)−B(g2) B(g2) · · · B(gr ) P and there do not exist L and P for which the corresponding Since for every polynomial p(y) we have g1 − g2 | p(g1) − minimal operator M has order rLrP and larger degree. p(g2), every entry of the first rL columns in the above matrix is divisible by g1 − g2. Hence, the whole determinant is rL divisible by (g1 − g2) . 4. ORDER-DEGREE-CURVE Theorem 8. The minimal operator M ∈ C[x][∂] annihi- BY SINGULARITIES lating f ◦ g for every f and g such that L(f) = 0 and A singularity of the minimal operator M is a root of its P (x,g(x))=0 has order r ≤ rLrP and degree at most leading coefficient polynomial lc∂ (M) ∈ C[x]. In the nota- 2 1 tion and terminology of [7], a factor p of this polynomial 2r dP − 2 (r−2)(r−1) + rdP rL(2rP +dL−1) − dP rL(rP −1) is called removable at cost n if there exists an operator = O(rdP rL(dL + rP )). Q ∈ C(x)[∂] of order deg∂ (Q) ≤ n such that QM ∈ C[x][∂] and gcd(lc∂ (QM),p) = 1. A factor p is called removable if Proof. We construct M using det Cℓ for 1 ≤ ℓ ≤ r + 1. it is removable at some finite cost n ∈ N, and non-removable We consider some f and by F we denote the (rLrP + 1)- otherwise. The following theorem [7, Theorem 9] translates (r) T dimensional vector (f,...,f , 0,..., 0) . If f ∈ V , then information about the removable singularities of a minimal the first r +1 rows of the matrix C F are linearly depen- operator into an order-degree curve. dent, so it is degenerate. On the other hand, if this matrix is degenerate, then Lemma 6 implies that F is a linear combi- Theorem 11. Let M ∈ C[x][∂], and let p1,...,pm ∈ C[x] nation of the columns of C, so Lemma 5 implies that f ∈ V . be pairwise coprime factors of lc∂ (M) which are removable at costs c1,...,cm, respectively. Let r ≥ deg∂ (M) and k. On the other hand, corank M(α) is equal to the dimension m of the space of pairs of polynomials (a(y),b(y)) such that + ci a(y)P (α, y)+ b(y)Py(α, y) = 0 and deg b(y) 400 µα ≤ rLπα + λβi . (4) i=1 300 · X c · We sum (4) over all α ∈ C¯. The number of occurrences of 200 · d ¯ · λβ in this sum for a fixed β ∈ C is equal to the number of ·· i ···· distinct power series of the form g(x)= β+ ci(x−γ) such 100 ················ · · · · e that P (x,g(x)) = 0. Inverting these power series, we obtain a r distinct Puiseux series solutions of P (x,y)=0P at y = β, so 100 200 300 this number does not exceed dP . Hence µα ≤ rL πα +dP λβ ≤ 2rLdP (2rP −1)+dP dL. 4.1 Degree of Removable Factors ¯ ¯ ¯ αX∈C αX∈C βX∈C Lemma 13. Let P (x,y) ∈ C[x,y] be a polynomial with In order to use Theorem 11, we need a lower bound for deg P = d, and R(x) = Resy(P, Py). Assume that α ∈ C¯ y deg qrem. Theorem 8 gives us an upper bound for degx M, is a root of R(x) of multiplicity k. Then the squarefree part but we must also estimate the difference degx M −deg lc∂ M. By Nα we denote the Newton polygon for M at α ∈ C¯ ∪{∞} S(y)= P (α, y) gcd P (α, y), Py(α, y) (for definitions and notation, see [11, Section 3.3]). By Hα, of P (α, y) has degree at least d − k. we denote the difference of the ordinates of the highest and the smallest vertices of Nα, and we call this quantity the Proof. Let M(x) be the Sylvester matrix for P (x,y) and height of the Newton polygon. Note that H∞ ≤ degx M − (k) Py(x,y) with respect to y. The value R (α) is of the form deg lc∂ M. This estimate together with the Lemma above det Mi(α), where every Mi(x) has at least 2d − 1 − k implies deg qrem ≥ degx(M) − H∞ − dP (4rLrP − 2rL + dL). (k) common columns with M(x). Since R (α) 6= 0, at least one The equation P (x,y) = 0 has rP distinct Puiseux series P of these matrices is nondegenerate. Hence, corank M(α) ≤ solutions g1(x),...,grP (x) at infinity. For 1 ≤ i ≤ rP , let βi = gi(∞) ∈ C¯ ∪ {∞}, and let ρi be the order of zero of Conditions (S1) and (S2) ensure that zero is not a po- 1 gi(x) − βi ( , resp.) at infinity if βi ∈ C¯ (βi = ∞, resp.). tential true singularity of M(L). Condition (G) is an es- gi(x) The numbers ρ1,...,ρrP are positive rationals and can be sential technical assumption on P . We note that it holds read off from Newton polygons of P (see [1, Chapter II]). at all nonsingular points (not just at zero) for almost all P , because this condition is violated at α iff some root of rP Lemma 15. H∞ ≤ ρiHβ . i=1 i P (α, y) = Px(α, y) = 0 (this means that at least one of βi Proof. Writing L as LP(x,∂) ∈ C[x][∂], we have is zero) is also a root of either Pxx(α, y) = 0 (then γi is also zero) or Pxy(α, y) = 0 (then there are at least two such β’s). 1 1 For a generic P this does not hold. M = lclm L g , ∂ ,...,L gr , ∂ . 1 g′ P g′ Under these assumptions we will prove the following the- 1 rP orem. Informally speaking, it means that if M(L) has an Hence, the set of edges of N∞ is a subset of the union of sets 1 apparent singularity at zero, then it almost surely is remov- of edges of Newton polygons of the operators L(gi, ′ ∂), so gi able at cost one. the height of N∞ is bounded by the sum of the heights of the Newton polygons of these operators. Consider g1 and Theorem 18. Let dL be such that dL ≥ (rLrP − rL + 1)rP . ¯ assume that β1 ∈ C¯. Then the Newton polygon for L at β1 By V we denote the (algebraic) set of all L ∈ C[x][∂] of order is constructed from the set of monomials of L written as an rL and degree ≤ dL such that the leading coefficient of L ˜ element of C(x−β1)[(x−β1)∂]. Let L(x,∂)= L(x−β1, (x− does not vanish at α1,...,αrP . We consider two (algebraic) β1)∂), then subsets in V 1 g1−β1 −ρ X = L ∈ V M(L) has an apparent singularity at 0 , L g , ∂ = L˜ g −β , ∂ = L˜ x 1 h (x), xh (x)∂ , 1 g′ 1 1 g′ 1 2 1 1 Y = L ∈ V M(L) has an apparent singularity at 0 where h (∞) and h (∞) are nonzero elements of C¯. Since 1 2 which is not removable at cost one . h1 and h2 do not affect the shape of the Newton polygon 1 Then, dim X > dim Y as algebraic sets. at infinity, the Newton polygon at infinity for L(g1, ′ ∂) is g1 ¯ obtained from the Newton polygon for L at β1 by stretching For α ∈ C, byOpα(r, d) we denote the space of differential it vertically by the factor ρ1, so its height is equal to ρ1Hβ1 . operators in C¯[x − α][∂] of order at most r and degree at ˜ 1 The case β1 = ∞ is analogous using L = L x , −x∂ . most d. By NOpα(r, d) ⊂ Opα(r, d) we denote the set of L such that ord L = r and (lc∂ L)(α) 6= 0. Then Remark 16. Generically, the βi’s will be ordinary points of L, so it is fair to expect H = 0 for all i in most situations. V ⊂ NOp (rL,dL) ∩ . . . ∩ NOp (rL,dL). βi α1 αrP The following theorem is a consequence of Theorem 11 To every operator L ∈ NOpα(r, d0) and d1 ≥ r, we as- and the discussion above. sign a fundamental matrix of degree d1 at α, denote it by Fα(L, d1). It is defined as the r × (d1 + 1) matrix such that Theorem 17. Let ρ1,...,ρrP be as above. Assume that all removable singularities of M are removable at cost at the first r columns constitute the identity matrix Ir, and ev- rP ery row consists of the first d1 +1 terms of some power series most c. Let δ = ρiHβ + dP (4rLrP − 2rL + dL). Let i solution of L at x = α. Since L ∈ NOpα(r, d0), F (L, d1) is i=1 well defined for every d1. r ≥ deg∂ M + c − 1Pand By F (r, d) we denote the space of all possible fundamen- c c d ≥ δ · 1 − + deg M · . tal matrices of degree d for operators of order r. This space x r(d+1−r) r − deg∂ (M) + 1 r − deg∂ (M) + 1 is isomorphic to A . The following proposition says Then there exists an operator Q ∈ C(x)[∂] such that QM ∈ that a generic operator has generic and independent funda- mental matrices, so we can work with these matrices instead C[x][∂] and deg∂ (QM)= r and degx(QM)= d. of working with operators. Note that degx(M) may be replaced with the expression rP from Theorem 8 or Conjecture 10. Proposition 19. Let ϕ: V → (F (rL,rLrP )) be the map sending L ∈ V to Fα1 (L, rLrP )⊕. . . ⊕FαrP (L, rLrP ). Then 4.2 Cost of Removable Factors ϕ is a surjective map of algebraic sets, and all fibers of ϕ The goal of this final section is to explain why in the case have the same dimension. rP > 1 one can almost always choose c = 1 in Theorem 17. For a differential operator L ∈ C[x][∂], by M(L) we denote For the proof we need the following lemma. the minimal operator M such that Mf(g(x)) = 0 whenever Lemma 20. Let ψ : NOp (r, d) → F (r, d + r) be the map Lf = 0 and P (x,g(x)) = 0. We want to investigate the pos- α sending L to Fα(L, d+r). Then ψ is surjective and all fibers sible behaviour of a removable singularity at α ∈ C when L have the same dimension. varies and P with rP > 1 is fixed. Without loss of generality, we assume that α = 0. Proof. First we assume that L is of the form L = ∂rL + We will assume that: rL−1 d arL−1(x)∂ + . . . + a0(x), and aj (x)= aj,dx + . . . + aj,0, where aj,i ∈ C¯. We also denote the truncated power series (S1) P (0,y) is a squarefree polynomial of degree rP ; corresponding to the j + 1-st row of F (L, d + rL) by fj and (S2) g(0) is not a singularity of L for any root g(x) of P ; write it as (G) Roots of P (x,g(x)) = 0 at zero are of the form gi(x)= d 2 αi + βix + γix + . . ., where β ,...,βr are nonzero, j rL+i 2 P fj = x + bj,ix , where bj,i ∈ C.¯ and either β1 or γ1 is nonzero. i=0 X We will prove the following claim by induction on i: Lemma 20 implies that ϕ2 is also surjective and all fibers Claim. For every 0 ≤ j ≤ rL − 1 and every 0 ≤ i ≤ d, are of the same dimension. bj,i can be written as a polynomial in ap,q with q < i and ¯ aj,i. And, vice versa, aj,i can be written as a polynomial in Let g1(x),...,grP (x) ∈ C[[x]] be solutions of P (x,y) = 0 bp,q with q < i and bj,i. at zero. Recall that gi(x)= αi + βix + . . . for all 1 ≤ i ≤ rP , The claim would imply that ψ defines an isomorphism of and by (G) we can assume that β2,...,βrP are nonzero. algebraic varieties between Fα(rP ,d + r) and the subset of Consider A ∈ F (rL,d), assume that its rows correspond ¯ monic operators in NOpα(r, d). to truncations of power series f1,...,frL ∈ C[[x − αi]]. By For i = 0, looking at the constant term of L(fj ), we obtain ε(gi,A) we denote the rL × (d + 1)-matrix whose rows are ¯ that j!aj,0 + rL!bj,0 = 0. This proves the base case of the truncations of f1 ◦ gi,...,frL ◦ gi ∈ C[[x]] at degree d + 1. induction. Now we consider i > 0 and look at the constant term of Lemma 21. We can write ε(gi,A)= A · T (gi), where T (gi) i i is an upper triangular (d+1)×(d+1)-matrix depending only ∂ L(fj ). The operator ∂ L can be written as d on gi with 1, βi,...,βi on the diagonal. i i+rL (i) rL−1 (i) 2 ∂ L = ∂ + ar −1(x)∂ + . . . + a0 (x) Futhermore, if βi = 0 and gi(x) = αi + γix + . . ., then L d+3 (k) l the i-th row of T (gi) is zero for i ≥ 2 , and starts with + ck,l,sas (x)∂ i−1 d+3 2(i − 1) zeroes and γi for i< 2 . k ϕ1 : Op (rL,dL) → Op (rL,d0) ⊕ . . . ⊕ Op (rL,d0). 0 α1 αrP • M(L) has order less than rLrP or has an apparent singularity at zero which is either not removable at This map is linear, so it is sufficient to show that the dimen- cost one or of degree greater than one iff both π(L) sion of the kernel is equal to the difference of the dimensions (1) and π(L) are degenerate. of the source space and the target space. The latter num- (2) ber is equal to (dL + 1)(rL + 1) − (d0 + 1)(rL + 1)rP . Let Let X0 = {L ∈ V | det π(L)(1) = 0} and Y0 = {L ∈ V | L ∈ ker ϕ1. This is equivalent to the fact that every coeffi- det π(L)(2) = 0}, then X0 \ Y0 ⊂ X ⊂ X0 and Y ⊂ Y0. d0+1 cient of L is divisible by (x − αi) for every 1 ≤ i ≤ rP . The dimension of the space of such operators is equal to Proposition 23. ϕ(X0) is an irreducible subset of W , and (rL + 1)(dL + 1 − rP (d0 + 1)) ≥ 0, so ϕ1 is surjective. ϕ(Y0) is a proper algebraic subset of ϕ(X0). Proof. The above discussion and the surjectivity of ϕ imply Remark 25. On the other hand, neither Theorem 18 nor that ϕ(X0) = {N ∈ W | det ε(N)(1) = 0}. Hence, we need our experiments support the choice c = 1 in the case rP = 1. to prove that det ε(N)(1) is a nonzero irreducible polynomial Instead, it seems that in this case the cost for removability is ¯ in R = C[X1,...,XrP ]. We set A = ε(N)(1). systematically larger. To see why, consider the special case We claim that there is a way to reorder columns and rows P = y − x2 of substituting the polynomial g(x)= x2 into a of A such that it will be of the form solution f of a generic operator L. If the solution space of L admits a basis of the form B C1 , rL rL+1 C2 D 1 + a1,r x + a1,r +1x + · · · , L L rL rL+1 where B and D are square matrices, and x + a2,rL x + a2,rL+1x + · · · , . • B is upper triangular with nonzero elements of C¯ on . the diagonal; rL−1 rL rL+1 x + arL−1,rL x + arL−1,rL+1x + · · · , • entries of D are algebraically independent over the sub- and M is the minimal operator for the composition, then its algebra generated in R by entries of B,C1, and C2. solution space obviously has the basis 2rL 2rL+2 In order to prove the claim we consider two cases: 1 + a1,rL x + a1,rL+1x + · · · , 2 2rL 2rL+2 x + a2,rL x + a2,rL+1x + · · · , 1. β1 6= 0. By Corollary 22, A is already of the desired form with B being an r × r -submatrix. . L L . 2 2(rL−1) 2rL 2rL+2 2. β1 = 0. Then (G) implies that g1(x)= α1 + γ1x + . . . x + arL−1,rL x + arL−1,rL+1x + · · · , with γ1 6= 0. Then Lemma 21 implies that the follow- ing permutations would give us the desired block struc- and so the indicial polynomial of M is λ(λ−2) · · · (λ−2(rL − 1)). According to the theory of apparent singularities [6, 5], ture with B being an ⌊3rL/2⌋×⌊3rL/2⌋-submatrix, for columns: M has a removable singularity at the origin and the cost of removability is as high as rL. 1, 3,..., 2rL − 1, 2, 4,..., 2⌊rL/2⌋, ∗, More generally, if g is a rational function and α is a root of g′, so that g(x)= c + O((x − α)2), a reasoning along the and for rows: same lines confirms that such an α will also be a removable singularity with cost rL. 1, 2,...,rL,rL + 2,rL + 4,...,rL + 2⌊rL/2⌋, ∗, Acknowledgement. We thank the referees for their con- where ∗ stands for all other indices in any order. structive critizism. Using elementary row operations, we can bring A to the 5. 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