Complete ideals and bivariate Hermite interpolation

Antonio Campillo^{∗} Departamento de Algebra, Geometría y Topología Universidad de Valladolid

Santiago Encinas^{†} Departamento de Matemática Aplicada Fundamental Universidad de Valladolid

Introduction.

Polynomial interpolation in one variable has a well known and good theory. For multivariate polynomials, and even in case of bivariate polynomials, the situation becomes more difficult and different phenomena appear (see [7]). Polynomial interpolation in the subspace K[X, Y]_{n} of polynomials of degree ≤ n corresponds to a linear map L : K[X, Y]_{n} ⟶ K^{N} (1.2). Where the coordinates of L are evaluation of polynomials, or some partial derivatives, at prescribed points of K^{2}. Lagrange interpolation corresponds to just evaluation of polynomials, Hermite interpolation add partial derivatives at some points till certain order, and Birkhoff interpolation evaluates selected derivatives. The interpolation problem consist in finding a preimage f ∈ K[X, Y]_{n} of any evaluated point b ∈ K^{N}. The above evaluation functions must be independent conditions in order to have L surjective, and we say that the interpolation is poised. For example in case of Lagrange interpolation if points lie in some curve of low degree then independence fail. We find two problems for bivariate polynomial interpolation: first to generate poised interpolation problems with independent conditions, and second once we have a poised problem, to find the interpolation polynomial for a given b. For the Lagrange case some constructions can be found in [1] related to both problems; in that paper Cayley-Bacharach theorem is used to generate interpolation sets of points with independent evaluation functions. One starts with a complete intersection I_{0} and an ideal I ⊃ I_{0}, which corresponds to an interpolation problem. If I′ is the complement of I in I_{0} then the interpolation problem of I is poised if and only if that of I′ is poised (2.2). In this paper we focus on Hermite interpolation. For this case we need a carefully local analysis at the nodes with multiplicity bigger than one. Such analysis is done by using theory and some results on complete ideals (2.4). In such a way that we may generate poised Hermite interpolation problems. The answer to the second problem will be treated in a forthcoming paper. For Lagrange interpolation, we only have two choose accurately the points, but in Hermite case, conditions in derivatives can be already dependent for any choice of points. It is well known that the Hermite problem, given by two different points P, Q ∈ K^{2} and the evaluation of the partial derivatives with order ≤ 1 is not poised for polynomial of degree ≤ 2. The six evaluations functions are not independent for any choice of P and Q. So that, not any configuration of the appropriate number of points and derivatives is allowed. We describe a procedure to construct poised Hermite interpolation problems (3.1), and we characterize the kind of Hermite configuration that we can achieve by this procedure (theorem 3.3). Not all the poised Hermite interpolation problems can be achieved by this procedure, as we illustrate in 3.6. Lemma 2.4 proves an equality for the integral closure of a complete intersection I_{0} and the conductor of a generic element of I_{0}. This lemma is used to get theorems 2.5 and 2.6, which gives to us formulas of recurrence. This recurrence formulas will be used in 3.1 to get our procedure. On the other hand, the problem of finding the interpolation polynomial f such that L(f) = b will be treated in a forthcoming paper.

∗Partially supported by Digicyt PB97-0471

†Partially supported by Digicyt PB96-0065

1 Bivariate interpolation.

1.1. In what follows K will be a field and K[X, Y] be the polynomial ring in two variables. We denote by K[X, Y]_{n} the subspace of polynomials of degree ≤ n. The dimension, as K-vector space of K[X, Y]_{n} is

d(n) = \binom{n + 2}{2}.

Note that although the field K is not assumed to be algebraically closed (it is very important in practice the case K = ℝ for instance), methods and tools used below are working out over the base field K. However, sometimes one should consider ideal supports or intersections in the plane which can have their physical status and coordinates over a finite extension of the ground field.

Definition 1.2. An interpolation problem in the subspace K[X, Y]_{n} is given a linear map L : K[X, Y]_{n} ⟶ K^{N} for some integer N. The interpolation problem is said to be poised if the map L is surjective and, hence N ≤ d(n).

If L is a map as above, the interpolation problem consists in the following: given a vector b ∈ K^{N}, find a polynomial f ∈ K[X, Y]_{n} such that L(f) = b. the condition to be poised means that the interpolation problem has a solution for every b ∈ K^{N}. If moreover N = d(n), then such solutions are unique. As examples, one has Lagrange interpolation when L is the evaluation vector at N different points of K^{2}, Hermite when L evaluates also derivatives till prescribed orders at concrete points, or Birkhoff when L evaluates selected derivatives.

Two of the main questions in polynomial interpolation are, on one hand, to construct sets of points for evaluating in Lagrange, Hermite or Birkhoff interpolation for N = d(n) conditions on K[X, Y]_{n} and being poised; and, on the other hand, to construct solutions to the concrete problems L(f) = b by an explicit formula or an easy computation. In our previous paper [1] we approached above two questions by using the Cayley-Bacharach theorem, in order to solve Lagrange interpolation. We refer to [8] for concepts and notations in algebraic geometry. To obtain similar results for the Hermite case, one would need a more careful local analysis at the nodes of multiplicity bigger than one. In this paper we will show how such analysis gives a solution for the fist question. Such solution can be done by using the theory and some properties of complete ideals. A solution for the second question will be treated in a forthcoming paper.

Definition 1.3. An interpolation set S is a set of points of K^{2} with multiplicities. So that every element of S is of the form (P, a_{P}), where P ∈ K^{2} and a_{P} is some non-negative integer. We denote by |S| the support of S, that is all the points P. To an interpolation set S we may associate an Hermite interpolation problem, say L : K[X, Y]_{n} ⟶ K^{N}, where

N = ∑_{P ∈|S|} \binom{a_{P} + 2}{2},

n is some integer, and L is the evaluation of all partial derivatives at points P ∈ |S| with order ≤ a_{P}.

We will focus our interest on constructing interpolation sets such that N = d(n) for some suitable n. Note also that if a_{P} = 0 for all P ∈ |S|, the interpolation set S correspond to a Lagrange interpolation and if some a_{P} > 0 we have an Hermite problem. Now, the notion we formalize the notion of multiplicity at points by means of zero dimensional ideals. So that one could define more general interpolation sets.

Definition 1.4. An interpolation ideal is an ideal I ⊂ K[X, Y] of finite colength together with a linear map L : K[x, y] ⟶ K^{N} with kernel equal to I. We recall that the colength of I, denoted by ℓ(I), can be computed as the dimension of the quotient K[x, y]/I as K-vector space. Note that we may consider interpolation problems (1.2) by considering the restriction map of L, which we also denote L, to K[X, Y]_{n} for suitable n.

The above interpolation problem is poised if and only if

dim_{K}(I ∩ K[X, Y]_{n}) = d(n) − N = \binom{n + 2}{2} − N

1.5. Given an interpolation set S (1.3) we may associate an interpolation ideal (1.4) I: The ideal I is such that the fiber at points P ∈ |S| is I_{P} = 𝔪_{P}^{a_{P} +1}, and at points Q ∉ |S|, I_{Q} = O_{𝔸^{2},P}.

Where O_{𝔸^{2}} is the structural sheaf of the 2-dimensional affine space over K, and 𝔪_{P} denotes the maximal ideal of the local ring O_{𝔸^{2},P}. And reciprocally, given an ideal with the conditions above, we may associate an interpolation set.

1.6. In order to apply algebraic geometry techniques we interpret the problem on the projective plane ℙ^{2} = ℙ_{K}^{2}. Now, the ideal I ⊂ K[X, Y] gives rise to a zero dimensional (coherent) sheaf I ⊂ O_{ℙ^{2}} with the same support and fibers as I. On the other hand, by homogenizing polynomials, K[X, Y]_{n} can be identified to the vector space H^{0}(O_{ℙ^{2}}(n)) and I ∩ K[X, Y]_{n} corresponds to H^{0}(I(n)) by such identification. Here, H^{0}(I(n)) is considered as a subspace of H^{0}(O_{ℙ^{2}}(n)) by means of the first map in the vector space long exact sequence in cohomology

0 ⟶ H^{0}(I(n)) ⟶ H^{0}(O_{ℙ^{2}}(n)) ⟶ H^{0}(O_{ℙ^{2}}/I(n)) ⟶ H^{1}(I(n)) ⟶ 0 (1.6.1)

induced by twisting the sheaf short exact sequence

0 ⟶ I ⟶ O_{ℙ^{2}} ⟶ O_{ℙ^{2}}/I ⟶ 0

By taking dimensions in 1.6.1 one gets the equality

h^{0}(I(n)) = h^{1}(I(n)) + d(n) − ℓ(I)  (1.6.2)

where ℓ(I) is the colength of the ideal sheaf I. It follows from the above equality that the interpolation is poised if and only if one has h^{1}(I(n)) = 0.

1.7. In order to calculate the value of h^{1}(I(n)), we use the Cayley-Bacharach theorem. Assume that the zero dimensional ideal I contains the global complete intersection ideal I_{0} of two (non necessarily reduced) projective plane curves D, E of respective degrees d, e and without common factors. Here I_{0} is the ideal sheaf supported at the points of D∩E and generated by the local equations of D and E at them. By Bezout theorem, the colength of I_{0} is de. Cayley-Bacharach theorem (see [6]) states that, if I′ = Ann(I/I_{0}), called the complement of I at I_{0}, then I′ is a new ideal sheaf containing I_{0} and one has

(i) the colength of I′ is ℓ(I′) = de − ℓ(I).

(ii) h^{1}(I(n)) = h^{0}(I′(d + e − 3 − n)) − h^{0}(I_{0}(d + e − 3 − n)).

We refer to [6] for details and more general statements. In particular, notice that I_{0} being locally complete intersection, property (i) is local. More specifically, if now ℓ denotes the colength of ideals at a local ring O_{ℙ^{2},Q}, one has ℓ(I′) = ℓ(I_{0})−ℓ(I), whenever I ⊃ I_{0}, I′ = Ann_{O}(I/I_{0}) and I_{0} is a primary ideal generated by two elements.

2 Main results.

Fix n ≥ 1 and in what follows we assume N = d(n) in all interpolation problems. So that for any interpolation ideal (1.4) with colength ℓ(I) = d(n), equality 1.6.2 looks as h^{0}(I(n)) = h^{1}(I(n)). By the discussion of 1.6 the interpolation is poised if and only if one has h^{0}(I(n)) = 0 or, in other words, if no degree n curve is passing though the scheme V (I). One deduces that the condition of being poised does not depends on L, but it only depends on I and n. This justifies the following definition:

Definition 2.1. An ideal sheaf I ⊂ O_{ℙ^{2}} is said to be an interpolation locus for K[X, Y]_{n} if one has ℓ(I) = d(n) and h^{0}(I(n)) = 0.

Proposition 2.2. Assume that I_{0} is the intersection scheme of two degree n+1 curves without common factor. Let I be an ideal sheaf containing I_{0} and I′ = Ann(I/I_{0}) be the complement of I relative to I_{0}. Then I is an interpolation locus (2.1) for K[X, Y]_{n} if and only if I′ is an interpolation locus for K[X, Y]_{n−1}.

Proof. First note that d(n)+d(n−1) = (n+1)^{2}. Thus, by 1.7(i) one has that ℓ(I) = d(n) if and only if ℓ(I′) = d(n − 1). Now, assume that ℓ(I) = d(n). We know also that h^{0}(I(n)) = h^{1}(I(n)) by 1.6.2. From 1.7(ii), one gets h^{1}(I(n)) = h^{0}(I′(n−1))−h^{0}(I_{0}(n−1)), Finally h^{0}(I_{0}(n−1)) = 0, since a nonzero form in H^{0}(I_{0}(n − 1)) would define a degree n − 1 curve whose intersection with a generic member of the pencil defined by the two curves intersecting at I_{0} will be of degree at least ℓ(I_{0}) = (n + 1)^{2} > (n + 1)(n − 1) contradicting Bezout theorem. Thus, one concludes h^{0}(I(n)) = 0 if and only if h^{0}(I′(n − 1)) = 0 as required. ⟲

Using this theorem, it is possible to construct Lagrange interpolation sets in a very easy way, see [1]. In fact, the main difficulties to apply it arise from the fact that the complement of some configurations of points of K^{2} relative to many possible choices of I_{0} are supported on points with coordinates in some finite extensions of K. This difficulty is avoided by taking intersecting curves which are a union of lines as is shown in [1] and as we will show again in 3.1. For the Hermite case the situation becomes much more complicated. To deal with it, we will need some properties of the local theory of complete ideals.

2.3. Let Q ∈ ℙ^{2} and denote by O the local ring O_{ℙ^{2},Q}. consider a primary ideal I_{0} ⊂ O. The integral closure, or completion, Ī_{0} is the ideal consisting of those elements h ∈ O satisfying an equality of type

h^{n} + a_{1}h^{n−1} + ⋯ + a_{n},  n ≥ 1, a_{i} ∈ I_{0}^{i}, i = 0, 1, …, n − 1  (2.3.1)

Associated to I_{0} there is the cluster of infinitely near base points (see [2] for definitions and details). This is a weighted tree K, called constellation in [2], of infinitely near points. If m_{P} is the weight of the point P, then I_{0} consists exactly of those h ∈ O such that the curve defined by h passes through all points with virtual multiplicity at least m_{P}.

An element h in I_{0} (or in Ī_{0}) is said to be a general element, if the corresponding curve passes through the points P with multiplicity exactly m_{P}. It follows that Ī_{0} can be generated by general elements. In other words, assume that Π : X̃ ⟶ ℙ^{2} denotes the composition of successive blow ups of the points in the cluster, and denote by E_{P}^{∗} the strict transform at X̃ of the exceptional divisor created by the blow up of the point P. If D = ∑_{P} E_{P}^{∗} one has that Ī_{0} is nothing but the fiber at Q of the ideal sheaf Π_{∗}OX̃(−D). The general elements of I_{0} are those h such that the exceptional part of the zero divisor Π_{∗}h is exactly D. Now the conductor (or adjoint ideal) C of I_{0} is defined to be the complete ideal given by the ideal sheaf Π_{∗}OX̃(−D′), where D′ = ∑(m_{P} − 1)E_{P}^{∗}. In other words, C is the set of functions, such that the curve which they define passes through the points P with virtual multiplicity at least m_{P} − 1. Thus the conductor is nothing but the lifting to O, by the map O ⟶ R_{f} = O/(f) of the conductor of the one dimensional local ring R_{f} for every general element f of I_{0}. Recall that the conductor of R_{f} is the ideal given by Ann(R̄_{f}/R_{f}), where R̄_{f} is the integral closure of R_{f} in its quotient ring.

Theorem 2.4. With the notations and assumptions of 2.3, assume that the ideal I_{0} is generated by two elements, such that it is locally complete intersection. Then we have that

Ī_{0} · C ⊂ I_{0}

Moreover, C = Ann_{O}(Ī_{0}/I_{0}) and Ī_{0} = Ann_{O}(C/I_{0})

Proof. Assume h ∈ Ī_{0} and take two general elements f, g ∈ I_{0} generating I_{0}. then h satisfies a relation as in 2.3.1. Here the a_{i} are homogeneous polynomials in f, g of degree i with coefficients in O.

Denote by h̃, g̃ the classes of h, g in R_{f}. It follows that \frac{h̃}{g̃} belongs to the integral closure

R̄_{f}, note that g̃ can not be a zero divisor in R_{f}. Now if c ∈ C and if c̃ denotes its class modulo f, one has c̃h̃ ∈ g̃R_{f} and hence c · k ∈ I_{0}. This proves formula CĪ_{0} ⊂ I_{0}. To show last equalities of theorem, we only need to take into account the well know Hoskin-Deligne formulas (see [9]):

ℓ(I_{0}) = ∑_{P} \frac{m_{P} (m_{P} + 1)}{2}

ℓ(C) = ∑_{P} \frac{m_{P} (m_{P} − 1)}{2}

and check that

ℓ(Ī_{0}) + ℓ(C) = ∑_{P} m_{P}^{2} = ℓ(I_{0})

this implies that Ī_{0} and C are the complement of each other. ⟲ Corollary 2.5. Let I_{0} be the intersection scheme of two curves C and D without common factor and with degrees d and e respectively. Let I be an ideal sheaf containing I_{0} and I′ = Ann(I/I_{0}) be the complement of I relative to I_{0}. Let Q be a point such that the multiplicities of C and D coincide and are equal to, say, r. If I_{Q} = 𝔪_{Q}^{r−j}, where j = 0, 1, then I′_{Q} = 𝔪_{Q}^{r−(1−j)}.

Proof. We know ℓ((I_{0})_{Q}) = r^{2}, ℓ(I_{Q}) = \binom{r−j+1}{2} and ℓ(I′_{Q}) = \binom{r−(1−j)+1}{2}, and note that

r^{2} = \binom{r − j + 1}{2} + \binom{r − (1 − j) + 1}{2}

It is enough to show that 𝔪_{Q}^{r(r−1)} ⊂ (I_{0})_{Q}, since then 𝔪_{Q}^{r−(1−j)} ⊂ I′_{Q} and both ideals have same colength. Now note that (I_{0})_{Q} ⊂ (Ī_{0})_{Q} ⊂ 𝔪_{Q}^{r}. By 2.4 and the fact that C ⊂ 𝔪_{Q}^{r−1} we obtain the result. ⟲

Corollary 2.6. Let I_{0} be the intersection scheme of two curves C and D without common factor and with degree n + 1. Assume also that C and D are union of lines. Let I be an ideal sheaf containing I_{0} and I′ = Ann(I/I_{0}) be the complement of I relative to I_{0}. Let Q be a point such that the multiplicity of C is r and the multiplicity of D is r + 1 (i. e. r component lines of C and r + 1 of D pass through Q). If I_{Q} = 𝔪_{Q}^{r}, then I′_{Q} = 𝔪_{Q}^{r}.

Proof. We have that ℓ((I_{0})_{Q}) = r(r + 1) and the fact that ℓ(I_{Q}) = \frac{1}{2}r(r + 1), implies

that ℓ(I′_{Q}) = \frac{1}{2}r(r + 1). Now note that it is enough to show 𝔪_{Q}^{2r} ⊂ (I_{0})_{Q}. If this holds, then 𝔪_{Q}^{r} ⊂ I′_{Q} and both ideals have same colength, so they must be equal. To see that 𝔪_{Q}^{2r} ⊂ (I_{0})_{Q}, consider the integral closure (Ī_{0})_{Q} of (I_{0})_{Q}. Recall that (Ī_{0})_{Q} can be computed by the cluster of points which consists of the point Q with multiplicity r and r points P_{1}, …, P_{r} in the exceptional divisor with multiplicity one, corresponding to the r directions of lines of C passing through Q. Then (Ī_{0})_{Q} consists of functions with multiplicity ≥ r at Q and such that the weak transform (with respect to r) vanish at P_{1}, …, P_{r}. In particular 𝔪_{Q}^{r+1} ⊂ (Ī_{0})_{Q}. On the other hand the conductor C of the generic element of (I_{0})_{Q} is C = 𝔪_{Q}^{r−1}. Now by 2.4 we conclude that 𝔪_{Q}^{2r} ⊂ (I_{0})_{Q}, as required. ⟲

3 Algorithm.

In this section we will compute interpolation locus (2.1) associated to Hermite interpolation sets (1.5). In order to apply theorem 2.2 we consider two curves of degree n + 1 which are union of lines. Also we want to ensure that the complement I′ of an Hermite problem I is again an Hermite problem, so that we must add some additional conditions to the ideal I.

3.1. Let L_{0}, …, L_{n} and M_{0}, …, M_{n} 2(n+1) distinct lines of ℙ^{2}. Set C(n) = L_{0} ∪⋯∪ L_{n} and D(n) = M_{0} ∪ ⋯ ∪ M_{n}. Assume that all points of the intersection C(n) ∩ D(n) are included in K^{2}. Set I_{0} the ideal of the intersection C(n) ∩ D(n) Let S(n − 1) be an interpolation set, such that

(1) |S(n − 1)| ⊂ C(n) ∩ D(n).

(2) The associated interpolation ideal I is an interpolation locus for K[X, Y]_{n−1} (2.1).

(3) For any (P, a_{P}^{(n−1)}) ∈ S(n − 1) then one of the following cases hold:

(a) ν_{P} (C(n)) = ν_{P} (D(n)) = a_{P}^{(n−1)} + 1 + j, with j = 0, 1.

(b) ν_{P} (C(n)) = a_{P}^{(n−1)} + 1 + j and ν_{P} (D(n)) = a_{P}^{(n−1)} + 1 + (1 − j), with j = 0, 1

Note that the multiplicity of C(n) or D(n) is computed by counting the number of lines which are components of the curve and pass through the point. Under this conditions, if I′ is the complement of I in I_{0}, it follows from 2.5 and 2.6 that I′ is an ideal as in 1.5 and there is an Hermite interpolation set S(n) associated to I′. Moreover by theorem 2.2 we have that I′ is an interpolation locus for K[X, Y]_{n}.

3.2. The construction of 3.1 allows to construct Hermite interpolation sets with associated interpolation locus. The procedure is to add two new lines L_{n+1} and M_{n+1} in such a way that the set S(n) will also satisfy conditions (1), (2) and (3) for the curves

C(n + 1) = L_{0} ∪ ⋯ ∪ L_{n} ∪ L_{n+1}

D(n + 1) = M_{0} ∪ ⋯ ∪ M_{n} ∪ M_{n+1}

which must be such that C(n + 1) ∩ D(n + 1) ⊂ 𝔸^{2}. Condition (1) is trivial, if we proceed by recurrence. Condition (2) will hold also by recurrence, since we always produce an interpolation locus. And condition (3) depends on the choice of the lines L_{n+1} and M_{n+1}.

Theorem 3.3. Let S(n) be as in 3.1, so that conditions (1), (2) and (3) hold, then S(n) is of the form:

S(n) = {(P_{1}, a_{1}), …, (P_{r}, a_{r}), (Q_{1}, 0), …, (Q_{s}, 0)}

where a_{i} > 0, i = 1, …, r, and

∑_{i=1}^{r}(a_{i} + 1) ≤ n + 1 s = d(n) − ∑_{i=1}^{r} d(a_{i})

Note that except case r = 1, always s > 0.

Proof. Set P_{i,j} = L_{i} ∩ M_{j}, i, j = 0, …, n and consider the matrix A = {P_{i,j}}_{i,j=0}^{n} with entries the above points of C(n) ∩ D(n). Not all the entries of A are different points, in fact if some components of C(n) and D(n) pass through the same point, all the entries of a submatrix of A coincide. By conditions (3.a) and (3.b) these submatrices must be of dimension (a_{P} + 1 + j_{1}) × (a_{P} + 1 + j_{2}), where j_{1}, j_{2} ∈ {0, 1}.

In the figure, P_{1} is a point belonging to 4 lines of C(n) and 4 lines of D(n), P_{2} is the intersection of 3 lines of C(n) and 3 of D(n) and P_{3} corresponds to 2 lines of one curve and 3 of the other one. Let A_{1} and A_{2} be two submatrices corresponding to two multiple points P_{1}, P_{2} ∈ C(n) ∩ D(n). There are sets of indices

Λ_{1} = {i_{1}, …, i_{r}} × {j_{1}, …, j_{s}}

Λ_{2} = {i′_{1}, …, i′_{r′}} × {j′_{1}, …, j′_{s′}}

such that

A_{1} = {P_{i,j}}_{(i,j)∈Λ_{1}}, A_{2} = {P_{i,j}}_{(i,j)∈Λ_{2}}

Now we may deduce that either

(i) {i_{1}, …, i_{r}} ∩ {i′_{1}, …, i′_{r′}} = ∅ and {j_{1}, …, j_{s}} ∩ {j′_{1}, …, j′_{s′}} = ∅, or

(ii) {i_{1}, …, i_{r}}∩{i′_{1}, …, i′_{r′}} has only one element, and {j_{1}, …, j_{s}}∩{j′_{1}, …, j′_{s′}} = ∅, or

(iii) {i_{1}, …, i_{r}}∩{i′_{1}, …, i′_{r′}} = ∅ and {j_{1}, …, j_{s}}∩{j′_{1}, …, j′_{s′}} has only one element.

Since otherwise the two points of C(n) ∩ D(n) are the same or we find the case of the following figure and we are considering lines. Now it is clear that a_{P_{1}} + 1 + a_{P_{2}} + 1 ≤ n + 1. If we consider all the submatrix corresponding to multiple points, we conclude that

∑_{i=1}^{r}(a_{i} + 1) ≤ n + 1

And the rest of non multiple points, Q_{1}, …, Q_{s}, must be enough to get d(n) conditions:

s = d(n) − ∑_{i=1}^{r} d(a_{i})

⟲

Corollary 3.4. Given multiplicities a_{1}, …, a_{n} and an integer n. An interpolation set for Hermite bivariate interpolation, say

S = {(P_{1}, a_{1}), …, (P_{r}, a_{r}), (Q_{1}, 0), …, (Q_{s}, 0)}

corresponding to an interpolation locus, can be constructed by the procedure of 3.1 if and only if one has

∑_{i=1}^{r}(a_{i} + 1) ≤ n + 1

Note that, as it would be expected, the interpolation set S = {(P, 1), (Q, 1)} for K[X, Y]_{2} is not achieved, since we have 4 > n + 1 = 3.

3.5. As an example, we use our procedure to obtain an interpolation set

S = {(P_{1}, 2), (P_{2}, 1), (Q_{1}, 0), (Q_{2}, 0), (Q_{3}, 0), (Q_{4}, 0), (Q_{5}, 0), (Q_{6}, 0)}

Start with six different lines passing trough the point P_{1}, say L_{0}, L_{1}, L_{2} and M_{0}, M_{1}, M_{2}. By 2.5 we obtain the interpolation set

S(2) = {(P_{1}, 2)}

Now add two lines L_{3} and M_{3} such that P_{1}, P_{2} ∈ L_{3} and P_{2} ∈ M_{3}, to produce

S(3) = {(P_{1}, 2), (P_{2}, 0), (R_{1}, 0), (R_{2}, 0), (R_{3}, 0)}

where R_{1}, R_{2}, R_{3} are the intersection points of M_{3} with L_{0}, L_{1}, L_{2}. Finally add two new lines L_{4} and M_{4} passing through P_{2} to get S = S(4).

In the figure, dashed lines are the M_{i} and continuous lines are the L_{1}.

3.6. We illustrate with an example that there are more Hermite interpolation sets corresponding to interpolation locus which are not like in theorem 3.3. The interpolation set

S = {(P_{1}, 2), (P_{2}, 1), (P_{3}, 1), (P_{4}, 1)}

where P_{1} = (0, 0), P_{2} = (1, 0), P_{3} = (0, 1) and P_{4} = (1, 1), corresponds to an ideal I which is an interpolation locus for K[X, Y]_{4}. But S is not like in theorem 3.3 since

(2 + 1) + (1 + 1) + (1 + 1) + (1 + 1) > n + 1 = 5

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[8] R. Harsthorne. Algebraic Geometry. GTM Springer-Verlag (1997).

[9] J. Lipman. On complete ideals in regular local rings. Alg. Geo. Comm. in Honour to Nagata, 203-231 (1987)

Antonio Campillo Dep. Álgebra Geometría y Topología Facultad de Ciencias Universidad de Valladolid Spain campillo@agt.uva.es

Santiago Encinas Dep. Matemática Aplicada Fundamental E.T.S. Arquitectura Universidad de Valladolid Spain sencinas@maf.uva.es