aa r X i v : . [ m a t h . C V ] J a n POLYNOMIAL INTERPOLATION AND RESIDUECURRENTS
JIMMY JOHANSSON
Abstract.
We show that a global holomorphic section of O ( d ) re-stricted to a closed complex subspace X ⊂ P n has an interpolant ifand only if it satisfies a set of moment conditions that involves a residuecurrent associated with a locally free resolution of O X . When X is afinite set of points in C n ⊂ P n this can be interpreted as a set of lin-ear conditions that a function on X has to satisfy in order to have apolynomial interpolant of degree at most d . Introduction
Let i : X ֒ → C n be a subvariety or complex subspace whose underlyingspace, X red , is a finite set of points { p , . . . , p r } ⊂ C n . Let g be a holo-morphic function on X , i.e., a global holomorphic section of O X , and let G ∈ C [ ζ , . . . , ζ n ] be a polynomial. We say that G interpolates g if thepull-back of G to X equals g , i.e., i ∗ G = g .If X is reduced, then a holomorphic function g on X is just a function from X red = { p , . . . , p r } to C , and we have that G interpolates g if G ( p j ) = g ( p j ) for each j = 0 , . . . , r . In the univariate case this is referred to as Lagrangeinterpolation. If X is not reduced, then at each point p j , G also has to satisfysome conditions on its derivatives. In the univariate case this is referred toas Hermite interpolation, see Example 4.3.The motivating question for this note is the following. What are thenecessary and sufficient conditions on g for the existence of an interpolantof degree at most d ?Let A X denote the vector space of holomorphic functions on X , i.e., A X = H ( C n , O X ) . Since the set of holomorphic functions on X that have aninterpolant of degree at most d is a linear subspace of A X , we have that afunction g ∈ A X has an interpolant of degree at most d if and only if itsatisfies a finite set of linear conditions. In this note we will show how theselinear conditions can be explicitly realized as a set of moment conditions thatinvolves a so-called residue current associated with a locally free resolutionof O X .Recall that since X red is a finite set of points, X can be viewed as a closedcomplex subspace of P n , and we have that polynomials of degree at most d Date : January 20, 2021.2010
Mathematics Subject Classification. on C n naturally correspond to global holomorphic sections of the line bundle O ( d ) → P n via d -homogenization. This motivates the following more generalnotion of interpolation that we shall consider in this note. Let i : X ֒ → P n be a closed complex subspace of arbitrary dimension. Let Φ and ϕ be globalholomorphic sections of O ( d ) and O X ( d ) = i ∗ O ( d ) , respectively. We saythat Φ interpolates ϕ if i ∗ Φ = ϕ .From a minimal graded free resolution of the homogeneous coordinate ringof X , S X , we obtain a locally free resolution of O X of the form −→ O ( E n ) f n −→ . . . f −→ O ( E ) f −→ O P n −→ O X −→ (1.1)where E k = L ℓ O ( − ℓ ) β k,ℓ , see [7] and [2, Section 6]. The β k,ℓ are referredto as the graded Betti numbers of S X . We equip the E k with the naturalHermitian metrics. In [2], Andersson and Wulcan showed that with (1.1), onecan associate a residue current R that generalizes the classical Coleff–Herreraproduct [5], see Section 2. It can be written as R = P k,ℓ R k,ℓ , where each R k,ℓ is an O ( − ℓ ) β k,ℓ -valued (0 , k ) -current. In [3], the same authors proved aresult which as a special case gives a cohomological condition in terms of thecurrent R for when Φ interpolates ϕ . In Section 3 we will show that in oursetting this condition amounts to the following set of moment conditions. Theorem 1.1.
Let X ⊂ P n be a closed complex subspace, and let R be theresidue current associated with (1.1). Moreover, let ω be a nonvanishingholomorphic O ( n + 1) -valued n -form. A global holomorphic section ϕ of O X ( d ) has an interpolant if and only if for each ℓ it holds that Z P n R n,ℓ ϕ ∧ hω = 0 (1.2) for all global holomorphic sections h of O ( ℓ − d − n − . Recall that the interpolation degree of X is defined as inf { d : all global holomorphic sections of O X ( d ) has an interpolant } . In particular, if X red is a finite set of points in C n , then the interpolationdegree of X is the smallest number d such that any g ∈ A X has an interpolantof degree at most d . Define t k ( S X ) = sup { ℓ : β k,ℓ = 0 } . (1.3)(We use the convention that the supremum of the empty set is −∞ .) As aconsequence of Theorem 1.1 we get the following bound of the interpolationdegree. Corollary 1.2.
Let X ⊂ P n be a closed complex subspace with homogeneouscoordinate ring S X . The interpolation degree of X is less than or equal to t n ( S X ) − n . It can be shown by purely algebraic means that the interpolation degreeof X is in fact equal to t n ( S X ) − n , see e.g. [9, Corollary 1.6]. If X red consistsof a finite set of points, then it can be shown that t n ( S X ) − n is equal to OLYNOMIAL INTERPOLATION AND RESIDUE CURRENTS 3 the Castelnuovo–Mumford regularity of S X , see [7, Exercise 4E.5], and thestatement in this case is Theorem 4.1 in [7].In Section 4 we will consider the case when X red is a finite set of points in C n . In this case the corresponding versions of Theorem 1.1 and Corollary 1.2first appeared in [12]. We will consider some examples where we explicitlywrite down the conditions for when g ∈ A X has an interpolant of degreeat most d . In particular, we will obtain the precise conditions for when theHermite interpolation problem has a solution.2. Residue currents
Let f be a holomorphic function in an open set in C n . Let ξ be a smooth ( n, n ) -form with compact support. In [8], using Hironaka’s desingularizationtheorem, Herrera and Lieberman proved that the limit lim ǫ → Z | f | >ǫ ξf (2.1)exists. Thus (2.1) defines a current known as the principal value current,which is denoted by [1 /f ] . The residue current R f of f is the (0 , -current ¯ ∂ [1 /f ] . It is easy to see that R f has its support on V ( f ) = f − (0) , and thatit satisfies the following duality principle : A holomorphic function Φ belongsto the ideal ( f ) if and only if R f Φ = 0 . Example . Let ζ ∈ C . We have that the action of ¯ ∂ [1 / ( ζ − ζ )] on a testform ξ ( ζ ) dζ is given by (cid:28) ¯ ∂ (cid:20) ζ − ζ (cid:21) , ξ ( ζ ) dζ (cid:29) = 2 πiξ ( ζ ) . (2.2) (cid:3) Residue currents associated with generically exact complexes.
We will now consider a generalization of the above construction due to An-dersson and Wulcan. Consider a generically exact complex of Hermitianholomorphic vector bundles over a complex manifold Y of dimension n , −→ E n f n −→ . . . f −→ E f −→ E −→ , (2.3)i.e., a complex that is exact outside an analytic variety Z ⊂ Y of positivecodimension. The vector bundle E = L k E k has a natural superbundlestructure, i.e., a Z -grading, E = E + ⊕ E − , where E + = L k E k and E − = L k E k +1 , which we shall refer to as the subspaces of even and odd elements,respectively. This induces a Z -grading on the sheaf of E -valued currents C ( E ) ; if ω ⊗ ξ is an E -valued current, where ω is a current and ξ is a smoothsection of E , then the degree of ω ⊗ ξ is the sum of the degree of ξ and thecurrent degree of ω modulo 2.We say that an endomorphism on E is even (resp. odd) if it preserves(resp. switches) the degree. If α is a smooth section of End E , then it defines JIMMY JOHANSSON a map on C ( E ) via α ( ω ⊗ ξ ) = ( − (deg α )(deg ω ) ω ⊗ α ( ξ ) , where ω is a current and ξ is a smooth section of E . In particular, the map f = P nk =1 f k defines an odd map on C ( E ) . We define an odd map on C ( E ) , ∇ = f − ¯ ∂ , which, since f and ¯ ∂ anti-commute, satisfies ∇ = 0 . The map ∇ extends to an odd map on C (End E ) via Leibniz’s rule, ∇ ( αξ ) = ( ∇ α ) ξ + ( − deg α α ∇ ξ. In [2], Andersson and Wulcan constructed
End E -valued currents U = X ℓ U ℓ = X ℓ X k ≥ ℓ +1 U ℓk , and R = X ℓ R ℓ = X ℓ X k ≥ ℓ +1 R ℓk , where U ℓk and R ℓk are Hom( E ℓ , E k ) -valued currents of bidegree (0 , k − ℓ − and (0 , k − ℓ ) , respectively, which satisfy ∇ U = id E − R, ∇ R = 0 . (2.4)The current R is referred to as the residue current associated with (2.3) andit has its support on Z .Suppose that the complex of locally free sheaves corresponding to (2.3), −→ O ( E n ) f n −→ . . . f −→ O ( E ) f −→ O ( E ) , (2.5)is exact. When the E k are equipped with Hermitian metrics, we shall referto (2.5) as a Hermitian resolution of the sheaf O ( E ) / im f . In this caseit holds that R ℓ = 0 if ℓ ≥ , and henceforth we shall write R k for R k .Moreover, we have that R satisfies the following properties: Duality principle : A holomorphic section Φ of E belongs to im f if andonly if R Φ = 0 . Dimension principle : If codim
Z > k , then R k = 0 .Note that the second equality in (2.4) is equivalent to f R = 0 , (2.6) f k +1 R k +1 − ¯ ∂R k = 0 , ≤ k ≤ n − , (2.7) ¯ ∂R n = 0 . (2.8)Let i : X ֒ → Y be a closed complex subspace with ideal sheaf I X , andsuppose that O X = i ∗ O Y , which we identify with O Y / I X , has a Hermitianresolution of the form −→ O ( E n ) f n −→ . . . f −→ O ( E ) f −→ O Y −→ O X −→ , (2.9)cf. (2.5) where E is the trivial line bundle. For the associated residuecurrent R = R + · · · + R n , we can view each R k as an E k -valued (0 , k ) -current. Since im f = I X , we have that i ∗ Φ = 0 if and only if R Φ = 0
OLYNOMIAL INTERPOLATION AND RESIDUE CURRENTS 5 by the duality principle. More generally, let L → Y be a holomorphic linebundle. If we equip L with a Hermitian metric, then we obtain a Hermitianresolution of i ∗ L = O X ⊗ L by tensoring (2.9) with L , and we have that R is the associated residue current with this resolution as well.2.2. The Coleff–Herrera product.
Let f = ( f , . . . , f p ) : C n → C p be aholomorphic mapping such that V ( f ) = f − (0) has codimension p . In [5]Coleff and Herrera gave meaning to the product µ f = ¯ ∂ (cid:20) f (cid:21) ∧ · · · ∧ ¯ ∂ (cid:20) f p (cid:21) , (2.10)which is known as the Coleff–Herrera product . In particular, if each f j onlydepends on ζ j , then (2.10) is just the tensor product of the one-variablecurrents ¯ ∂ [1 /f j ] described above. The current µ f is ¯ ∂ -closed, has support V ( f ) , and is anti-commuting in the f j . Moreover, µ f satisfies the dualityprinciple, i.e., µ f Φ = 0 if and only if Φ ∈ I ( f ) , where I ( f ) is the idealsheaf generated by f .Let H → Y be a holomorphic Hermitian vector bundle of rank p , andlet f be a holomorphic section of the dual bundle H ∗ . Let E k = V k H ,and define δ k : E k → E k − as interior multiplication by f . This gives agenerically exact complex (2.3). Suppose f = f e ∗ + · · · + f p e ∗ p in some localholomorphic frame e ∗ j for H ∗ . If codim f − (0) = p , then the correspondingcomplex of sheaves is a Hermitian resolution of O Y / I ( f ) known as the Koszul complex , and it was proven in [1] that the associated residue currentis given by R = R p = µ f e ∧ · · · ∧ e p .2.3. A comparison formula for residue currents.
We have the follow-ing comparison formula for residue currents, see Theorem 1.3 and Corol-lary 4.7 in [10]. Let X ⊂ X ′ be complex subspaces of codimension p of Y .Suppose that there exist Hermitian resolutions of length p of O X and O X ′ ,respectively, and let R and R ′ be the associated residue currents. Moreover,suppose that there exists a map of complexes O ( E ′ p ) · · · O ( E ′ ) O Y O X ′ O ( E p ) · · · O ( E ) O Y O X ψ p f ′ p f ′ ψ f ′ id f p f f Then R p = ψ p R ′ p .3. Interpolation and residue currents
Let Y be a complex manifold of dimension n , and let i : X ֒ → Y be aclosed complex subspace. Let L → Y be a holomorphic line bundle, and let Φ and ϕ be global holomorphic sections of L and i ∗ L , respectively. We saythat Φ interpolates ϕ if i ∗ Φ = ϕ . JIMMY JOHANSSON
Suppose that there exists a Hermitian resolution of O X of the form (2.9),and let R denote the associated residue current. For each point x ∈ Y thereis a neighborhood U and a holomorphic section e ϕ of L such that i ∗ e ϕ = ϕ on U . We define the current Rϕ on Y locally as R e ϕ . This is well-definedsince if e ϕ ′ is another section such that i ∗ e ϕ ′ = ϕ , then R ( e ϕ − e ϕ ′ ) = 0 by theduality principle.We have the following result which follows immediately as a special caseof Lemma 4.5 (ii) in [3]. Lemma 3.1.
Let Φ and ϕ be global holomorphic sections of L and i ∗ L ,respectively. Then Φ interpolates ϕ if and only if there exists a current w such that Φ − Rϕ = ∇ w . In other words, ϕ has an interpolant if and only if there exist currents w , . . . , w n such that ¯ ∂w n = R n ϕ , and ¯ ∂w k = f k +1 w k +1 + R k ϕ, ≤ k ≤ n − . (3.1)Moreover, in this case an interpolant of ϕ is given by Φ = f w . Note that Φ is holomorphic since ¯ ∂ Φ = − f ¯ ∂w = − f ( f w + R ϕ ) = − ( f R ) ϕ = 0 .Here the last equality follows from (2.6).Let us now consider interpolation on Y = P n with respect to the linebundle L = O ( d ) . Recall that there is a Hermitian resolution of O X of theform (1.1). We write R k,ℓ for the O ( − ℓ ) β k,ℓ -valued component of R k .If R n ϕ = ¯ ∂w n for some current w n , then one can successively find currents w n − , . . . , w such that (3.1) holds since, in view of (2.7), ¯ ∂ ( f k +1 w k +1 + R k ϕ ) = − f k +1 ¯ ∂w k +1 + ( ¯ ∂R k ) ϕ = − ( f k +1 R k +1 − ¯ ∂R k ) ϕ = 0 , and it follows from, e.g., [6, Theorem 10.7] that H ,k ( P n , E k ⊗ O ( d )) = 0 , ≤ k ≤ n − . We thus have the following condition for the existence of an interpolant.
Lemma 3.2.
A global holomorphic section ϕ of O X has an interpolant if andonly if R n ϕ is ¯ ∂ -exact, i.e., there exists a current η such that R n ϕ = ¯ ∂η .Proof of Theorem 1.1. By Serre duality we have that R n,ℓ ϕ is ¯ ∂ -exact if andonly if Z P n R n,ℓ ϕ ∧ η = 0 for all global ¯ ∂ -closed O ( ℓ − d ) -valued ( n, -forms η . Note that each suchform is of the form hω for some global holomorphic section h of O ( ℓ − d − n − . Since R n ϕ is ¯ ∂ -exact if and only if each component R n,ℓ ϕ is, thestatement follows from Lemma 3.2. (cid:3) Proof of Corollary 1.2.
Let d ≥ t n ( S X ) − n , see (1.3), and let ϕ be a globalholomorphic section of O X ( d ) . We have for each ℓ that Z P n R n,ℓ ϕ ∧ hω = 0 OLYNOMIAL INTERPOLATION AND RESIDUE CURRENTS 7 for all global holomorphic sections h of O ( ℓ − d − n − . Indeed, if ℓ ≤ d + n ,the only such h is the zero section, and if ℓ > d + n , then R n,ℓ = 0 since β n,ℓ = 0 . Therefore ϕ has an interpolant by Theorem 1.1. (cid:3) Polynomial interpolation
Let us now return to the topic of polynomial interpolation. Recall that thesetting is that X is a complex subspace of C n such that X red is a finite set ofpoints. The aim of this section is to give some examples where we explicitlycompute the residue current R associated with a Hermitian resolution of O X and write down the moment conditions that Theorem 1.1 imposes on afunction g ∈ A X for the existence of an interpolant of degree at most d . Wedo this by identifying g with a global holomorphic section ϕ of O X ( d ) anduse the fact that g has an interpolant of degree at most d if and only if ϕ hasan interpolant. More precisely, we let [ z ] = [ z : . . . : z n ] denote homogeneouscoordinates on P n , and we view C n as an open complex subspace of P n viathe embedding ( ζ , . . . , ζ n ) [1 : ζ : . . . : ζ n ] . Recall that on C n there is aframe e for O (1) such that a global holomorphic section Φ of O ( d ) is givenby Φ( ζ , . . . , ζ n ) = G ( ζ , . . . , ζ n ) e ( ζ , . . . , ζ n ) ⊗ d , where G is a polynomial of degree at most d on C n .Throughout this section we shall let ω in Theorem 1.1 be the nonvanishingholomorphic O ( n + 1) -valued n -form on P n such that ω = dζ ∧ · · · ∧ dζ n ⊗ e ⊗ ( n +1) on C n ⊂ P n .Note that the dimension principle gives that R = R n , and throughout thissection we write R ℓ rather than R n,ℓ for the O ( − ℓ ) β n,ℓ -valued component of R n . Example . Let X = { (0 , , (1 , , (0 , , (1 , } ⊂ C ⊂ P . We have that X is defined by the homogeneous ideal I X = ( f , f ) , where f = z ( z − z ) and f = z ( z − z ) . A Hermitian resolution of O X is given by the Koszulcomplex, see Section 2.2, where we interpret ( f , f ) as a global holomorphicsection of O (2) . Thus the associated residue current takes values in O ( − ,and is given by the Coleff–Herrera product, see Section 2.2, R = R = ¯ ∂ (cid:20) ζ ( ζ − (cid:21) ∧ ¯ ∂ (cid:20) ζ ( ζ − (cid:21) e ⊗ ( − . By a straightforward computation, cf. (2.1), we get R = (cid:18) ¯ ∂ (cid:20) ζ (cid:21) ∧ ¯ ∂ (cid:20) ζ (cid:21) − ¯ ∂ (cid:20) ζ − (cid:21) ∧ ¯ ∂ (cid:20) ζ (cid:21) − ¯ ∂ (cid:20) ζ (cid:21) ∧ ¯ ∂ (cid:20) ζ − (cid:21) + ¯ ∂ (cid:20) ζ − (cid:21) ∧ ¯ ∂ (cid:20) ζ − (cid:21)(cid:19) e ⊗ ( − . JIMMY JOHANSSON
By Theorem 1.1, we now get the following. Since R ℓ = 0 for ℓ ≥ , we havethat any g ∈ A X has an interpolant of degree at most . Moreover, g hasan interpolant of degree at most 1 if and only if (1.2) holds when ℓ = 4 and h = 1 . In view of (2.2) this amounts to g (0 , − g (1 , − g (0 ,
1) + g (1 ,
1) = 0 , (4.1)which is expected since the values of g at (0 , , (1 , , and (0 , uniquelydetermines a polynomial of degree at most 1 that takes the value g (1 ,
0) + g (0 , − g (0 , at (1 , . Note that this gives that the interpolation degreeof X is 2.We have that g has a constant interpolant if and only if (1.2) holds forall global holomorphic h of O (1) . By linearity we only need to check h = z , z , z , which amounts to (4.1), g (1 , − g (1 ,
0) = 0 , and g (1 , − g (0 ,
1) =0 . This amounts to g (0 ,
0) = g (1 ,
0) = g (0 ,
1) = g (1 , as expected. (cid:3) Example . Let X = { (0 , , (1 , , (0 , , (0 , } ⊂ C ⊂ P . We havethat X is defined by the homogeneous ideal I X = ( z a , z z , z a ) , where a = z − z and a = ( z − z )( z − z ) . We have Hermitian resolutions of O P / I ( z a , z a ) and O X and a map of complexes: O ( − O ( − ⊕ O ( − O P O P / I ( z a , z a ) O ( − ⊕ O ( − O ( − ⊕ O ( − O P O Xψ δ ψ δ id f f where the upper complex is the Koszul complex, see Section 2.2. Moreover, f = (cid:2) z a z z z a (cid:3) , f = − z a − a z , and ψ = , ψ = (cid:20) a a (cid:21) . Let R and R ′ denote the residue currents associated with the resolutions of O X and O / I ( z a , z a ) , respectively. We have that R ′ = R ′ takes valuesin O ( − and is given by the Coleff–Herrera product, see Section 2.2, R ′ = ¯ ∂ (cid:20) ζ ( ζ − (cid:21) ∧ ¯ ∂ (cid:20) ζ ( ζ − ζ − (cid:21) e ⊗ ( − . OLYNOMIAL INTERPOLATION AND RESIDUE CURRENTS 9
Thus by the comparison formula, see Section 2.3, R = ψ R ′ = R ⊕ R . Astraightforward computation gives that R = ¯ ∂ (cid:20) ζ ( ζ − (cid:21) ∧ ¯ ∂ (cid:20) ζ (cid:21) e ⊗ ( − = (cid:18) − ¯ ∂ (cid:20) ζ (cid:21) ∧ ¯ ∂ (cid:20) ζ (cid:21) + ¯ ∂ (cid:20) ζ − (cid:21) ∧ ¯ ∂ (cid:20) ζ (cid:21)(cid:19) e ⊗ ( − , and R = ¯ ∂ (cid:20) ζ (cid:21) ∧ ¯ ∂ (cid:20) ζ ( ζ − ζ − (cid:21) e ⊗ ( − = (cid:18)
12 ¯ ∂ (cid:20) ζ (cid:21) ∧ ¯ ∂ (cid:20) ζ (cid:21) − ¯ ∂ (cid:20) ζ (cid:21) ∧ ¯ ∂ (cid:20) ζ − (cid:21) + 12 ¯ ∂ (cid:20) ζ (cid:21) ∧ ¯ ∂ (cid:20) ζ − (cid:21)(cid:19) e ⊗ ( − . By Theorem 1.1, we now get the following. Since R ℓ = 0 for ℓ ≥ , we havethat any g ∈ A X has an interpolant of degree at most . Moreover, g hasan interpolant of degree at most 1 if and only if (1.2) holds when ℓ = 4 and h = 1 . (Note that there is no condition involving R since ℓ − d − n − < in this case.) In view of (2.2) we get the condition g (0 , − g (0 ,
1) + 12 g (0 ,
2) = 0 , (4.2)which is expected since g has an interpolant of degree at most 1 if and onlyif g (0 , is the average of g (0 , and g (0 , . Note that this gives that theinterpolation degree of X is 2.We get that g has a constant interpolant if and only if (1.2) holds when ℓ = 4 for h = z , z , z , and when ℓ = 3 and h = 1 . Since z vanishes on thesupport of R , this amounts to the equations (4.2), g (0 , − g (0 ,
1) = 0 and g (1 , − g (0 ,
0) = 0 . This amounts to g (0 ,
0) = g (1 ,
0) = g (0 ,
1) = g (0 , as expected. (cid:3) We end this note by considering Hermite interpolation. We refer to, e.g.,[4, 11, 13], and references therein for a classical survey of this topic.
Example . Let p , . . . , p r ∈ C , and let g be a holomorphic function on thecomplex subspace X ⊂ C defined by the ideal generated by Q rj =0 ( ζ − p j ) .Here we allow for the possibility that p i = p j for some i, j , so that X isnonreduced in general, and we denote the number of times that p j occursby m j . We have that a polynomial G interpolates g if and only if, for each j = 0 , . . . , r , G ( k ) ( p j ) = g ( k ) ( p j ) , k = 0 , . . . , m j − . We say that a polynomial interpolates g with respect to p , . . . , p k , k ≤ r ,if it interpolates the pull-back of g to the complex subspace defined by theideal generated by Q kj =0 ( ζ − p j ) . We denote the unique polynomial thatinterpolates g with respect to p , . . . , p k by H [ g ; p , . . . , p k ] . The coefficient of its ζ k -term is referred to as the k th divided difference of g and we denoteit by g [ p , . . . , p k ] . By induction it is not difficult to see that H [ g ; p , . . . , p r ]( ζ ) = r X k =0 g [ p , . . . , p k ] k − Y j =0 ( ζ − p j ) , see [4, Theorem 1.8]. This is referred to as Newton’s formula.We claim that g ∈ A X has an interpolant of degree at most d if and onlyif ( gh )[ p , . . . , p r ] = 0 for all polynomials h of degree at most r − d − . Letus show how this condition follows from Theorem 1.1. Since the ideal isgenerated by a single element, we have that the associated residue currentis given by R r +1 = ¯ ∂ " Q rj =0 ( ζ − p j ) e ⊗ ( − r − . Theorem 1.1 together with Stokes’ formula gives that g has an interpolantif and only if Z C ¯ ∂ " Q rj =0 ( ζ − p j ) gh ∧ dζ = Z C R H [ gh ; p , . . . , p r ]( ζ ) Q rj =0 ( ζ − p j ) dζ = 0 for all polynomials h of degree at most r − d − , where C R is a circle ofradius R ≫ . Here we have used the fact that H [ gh ; p , . . . , p r ] interpolates gh . By letting R → ∞ , a direct calculation gives that the second integral isequal to πi · ( gh )[ p , . . . , p r ] , and hence the claim follows. Acknowledgments
I would like to thank Elizabeth Wulcan and Mats Andersson for helpfuldiscussions and comments on preliminary versions of this note.
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