BIVARIATE LAGRANGE INTERPOLATION AT THE PADUA POINTS: THE IDEAL THEORY APPROACH

BIVARIATE LAGRANGE INTERPOLATION AT THE PADUA POINTS: THE IDEAL THEORY APPROACH LEN BOS, STEFANO DE MARCHI, MARCO VIANELLO, AND YUAN XU Abstract The Padua points are a family of points on the square [, ] given by explicit formulas that admits unique Lagrange interpolation by bivariate polynomials Interpolation polynomials and cubature formulas based on the Padua points are studied from an ideal theoretic point of view, which leads to the discovery of a compact formula for the interpolation polynomials The L p convergence of the interpolation polynomials is also studied Introduction For polynomial interpolation in one variable, the Chebyshev points (zeros of Chebyshev polynomial) are optimal in many ways Let L n f be the n-th Lagrange interpolation polynomial based on the Chebyshev points The Lebesgue constant L n in the uniform norm on [,] has growth of order of O(log n), which is the minimal rate of growth of any projection operator from C[,] onto the space of polynomials of degree at most n The Chebyshev points are also the nots of the Gaussian quadrature formula for the weight function / x on [,] In the case of two variables, it is often difficult to even give explicit point sets for which the polynomial interpolation problem is well-posed, or unisolvent Let Π n denote the space of polynomials of degree at most n in two variables It is well nown that dim Π n = (n + )(n + )/ Then consider a set V of points in R of cardinality V = dim Π n The set V is said to be unisolvent if for any given function f defined on R, there is a unique polynomial P such that P(x) = f(x) on V The Padua points are a family of interpolation points that may legitimately be considered as an analogue of the Chebyshev points for the square [,] For each n 0, they are defined by () Pad n = {x,j = (ξ,η j ), 0 n, j n + + δ }, where δ := 0 if n is even or n is odd but is even, δ := if n is odd and is odd, and () ξ = cos π cos j n, η n+ π, even j = cos j n+ π, odd Date: February 8, 007 99 Mathematics Subject Classification 4A05, 4A0 Key words and phrases bivariate Lagrange interpolation, Padua points, polynomial ideal The second and the third authors were supported by the ex-60% funds of the University of Padua and by the INdAM GNCS (Italian National Group for Scientific Computing) The fourth author was partially supported by NSF Grant DMS-0604056

LEN BOS, STEFANO DE MARCHI, MARCO VIANELLO, AND YUAN XU Padua Points for n=6 08 06 04 0 0 0 04 06 08 05 0 05 Figure It is easy to verify that the cardinality of Pad n is equal to the dimension of Π n These points were introduced heuristically in [4] (only for even degrees) and proved to be unisolvent in [3] The Lebesgue constant of the Lagrange interpolation based on the Padua points in the uniform norm grows lie O((log n) ), which is the minimal order of growth for any set of interpolation points in the square Moroever, there is a cubature formula of degree n based on these points for the weight function /( x y ) on [,] ([3]) A family of points studied in [0] also possesses similar properties ([,, 0]), but the cardinality of the set is not equal to dim Π n and the interpolation polynomial belongs to a subspace of Π n The study in [3] starts from the fact that the Padua points there lie on a single generating curve, γ n (t) = (cos(nt),cos((n + )t), 0 t π In fact, the points are exactly the distinct points among {γ n ( π n(n+) ), 0 n(n + )} In the present paper, we study the interpolation on the Padua points using an ideal theoretic approach, which considers the points as the variety of a polynomial ideal and it treats the cubature formula based on the interpolation points simultaneously The advantage of this approach is that it casts the result on Padua points into a general theoretic framewor The study in [0] used such an approach In particular, it shows how the compact formula of the interpolation polynomial arises naturally The order of the Lebesgue constant of the interpolation is found in [3], which solves the problem about the convergence of the interpolation polynomials on the

BIVARIATE LAGRANGE INTERPOLATION AT PADUA POINTS 3 Padua points in the uniform norm In the present paper, we show the convergence in a weighted L p norm The paper is organized as follows In the following section, we review the necessary bacground and show that the Padua points are unisolvent The explicit formula of the Lagrange interpolation polynomials is derived in Section 3, and the L p convergence of the interpolation polynomials is proved in Section 4 Polynomial ideal and Padua points Polynomial ideals, interpolation, and cubature formulas We first recall some notation and results about polynomial ideals and their relation to polynomial interpolation and cubature formulas To eep the notation simple we shall restrict to the case of two variables, even though all results in this subsection hold for more than two variables Let I be a polynomial ideal in R[x] with x = (x,x ), the ring of all polynomials in two variables If there are polynomials f,,f r in R[x] such that every f I can be written as f = a f ++a r f r, a j R[x], then we say that I is generated by the basis {f,,f r } and we write I = f,,f r Fix a monomial order, say the lexicographical order, and let LT(f) denote the leading term of the polynomial f in the monomial order Let LT(I) denote the ideal generated by the leading terms of LT(f) for all f I \ {0} Then it is is well nown that there is an isomorphism between R[x]/I and the space S I := span{x x j : x x j / LT(I) ([5, Chapt 5]) The codimension of the ideal is defined by codim(i) := dim(r[x]/i) For an ideal I of R[x], we denote by V = V (I) its real affine variety, ie, V (I) = {x R : p(x) = 0, p I} We consider the case of a zero dimensional variety, that is, when V is a finite set of distinct points in R In this case, it is well-nown that V codim I The following result is proved in [] Proposition Let I be a polynomial ideal in R[x] with finite codimension and let V be its affine variety If V = codim(i) then there is a unique interpolation polynomial in S I based on the points in the variety, ie, for every ordinate function z : V R there exists a unique p S I such that p(x) = z(x), x V In this case I coincides with the polynomial ideal I(V ), which contains all polynomials in R[x] that vanish on V An especially interesting case is when the ideal is generated by a sequence of quasi-orthogonal polynomials Let dµ be a nonnegative measure with finite moments on a subset of R A polynomial P R[x] is said to be an orthogonal polynomial with respect to dµ if R P(x)Q(x)dµ(x) = 0, Q R[x], deg Q < deg P; it is called a (n )-orthogonal polynomial if the above integral is zero for all Q R[x] such that deg P + deg Q n In particular, if P is of degree n + and (n ) orthogonal, then it is orthogonal with respect to all polynomials of degree n or lower We need the following result (cf []) Proposition Let I and V be as in the previous proposition Assume that I is generated by (n )-orthogonal polynomials If codim I = V then there is a cubature formula f(x)dµ(x) λ x f(x) R x V

4 LEN BOS, STEFANO DE MARCHI, MARCO VIANELLO, AND YUAN XU of degree n ; that is, the above formula is an identity for all f Π n The cubature formula in this proposition can be obtained by integrating the interpolation polynomial in Proposition We will apply this result for dµ = W(x)dx, where W is the Chebyshev weight function on [,], () W(x) = π x x, < x,x < For such a weight function, the cubature formula of degree n exists only if V satisfies Möller s lower bound V dim Π n + [n/] Formulas that attain this lower bound exist ([7, 8]), whose nots are the interpolation points studied in [0] Recall that the Chebyshev polynomials T n and U n of the first and the second ind are given by T n (x) = cos nθ and U n (x) = sin(n+)θ sin θ, x = cos θ, respectively The polynomials T n (x) are orthogonal with respect to / x Consequently, it is easy to verify that the polynomials () T (x )T n (x ), 0 n, are mutually orthogonal polynomials of degree n with respect to the weight function W(x,x ) on [,] Polynomial ideals and Padua points The Padua points () were introduced in [4] (apart from a misprint that n should be replaced by n + ) based on a heuristic argument that we now describe Morrow and Patterson [7] introduced, for s, an explicit set of nots in [,] for a cubature formula of degree s with respect to the weight function x x on [,] These Morrow-Patterson points are the common zeros of (3) R s j(x,x ) := U j (x )U s j (x ) + U s j (x )U j (x ), 0 j s There turn out to be exactly dim Π s many such points, and they are all in the interior (,) Moreover, they may be seen to lie on a certain collection of vertical and horizontal lines, forming a subset of a certain rectangular grid The Padua points of degree n are precisely the Morrow-Patterson points of degree n, together with the natural boundary points of this grid Figure shows the 8 Padua points of degree 6, consisting of the 5 interior Morrow-Patterson points of degree 6-=4, together with 3 additional boundary points It should be noted that the Morrow-Patterson points turn out to be sub-optimal in the sense that the associated Lebesgue constants grow faster than best possible In contrast, the Padua points have Lebesgue constants of optimal growth See ([]) for a discussion of this fact As shown in [3] the Padua points can be characterized as self-intersection points (interior points) and the boundary contact points of the algebraic curve T n (x ) + T n+ (x ) = 0, (x,x ) [,] (the generating curve), where T n denotes the n-th Chebyshev polynomial of the first ind (Actually the generating curve used in [3] is T n+ (x ) T n (x ) = 0) Depending on the orientation of the x and x directions, there are four families of Padua points, which are the two mentioned above and two others whose generating curves are T n (x ) T n+ (x ) = 0 and T n+ (x ) + T n (x ) = 0, respectively We restrict to the family () in this paper Theorem 3 Let Q n+ Π n+ be defined by (4) Q n+ 0 (x,x ) = T n+ (x ) T n (x ),

BIVARIATE LAGRANGE INTERPOLATION AT PADUA POINTS 5 and for n +, (5) Q n+ (x,x ) = T n + (x )T (x ) + T n + (x )T (x ) Then the set Pad n in () is the variety of the ideal I = Q n+ 0,Q n+,,q n+ n+ Furthermore, codim(i) = Pad n = dim Π n Proof That the polynomials Q n+ vanish on the Padua points () can be easily verified upon using the following representation and, for n +, Q n+ 0 (x,x ) = ( x )U n (x ) = sin θ sin nθ, Q n+ (x,x ) = cos ( )θ(cos (n + )φ cos φ + sin(n + )φ sin φ) + cos φ(cos nθ cos ( )θ + sinnθ sin ( )θ), where x = cos θ and x = cos φ Furthermore, the definition of Q n+ shows easily that the set {LT(Q n+ 0 ),,LT(Q n+ n+ )} is exactly the set of monomials of degree n + in R[x] Hence, it follows that dim R[x]/I dim Π d n Recall that codim(i) = dim R[x]/I V and V Pad n = dim Π d n, we conclude that codim(i) = Pad n The basis (4) and (5) of the ideal was originally identified by using the fact that the interior points of Pad n are the common zeros of (3) This implies that the ideal contains the polynomials ( x ) ( x ) R n j, 0 j n, as well as two specific univariate polynomials, ( x )U n (x ), which gives rise to (4), and a certain other polynomial in x (the details of which are not important) From these polynomials we were able to derive a simple basis for the ideal, the Q n+ Once the basis is identified, it is of course, after the fact, easier to simply verify it directly, as we did in the proof Since the product Chebyshev polynomials () are orthogonal with respect to W in (), it follows readily that Q n+ (x,x ) are orthogonal to the polynomials in Π n with respect to W on [,] In particular, they are (n )-orthogonal polynomials Hence, as a consequence of the theorem and Propositions and, we have the following: Corollary 4 The set of Padua points Pad n is unisolvent for Π d n Furthermore, there is a cubature formula of degree n based on the Padua points in Pad n We denote the unique interpolation polynomial based on Pad n by L n f, which can be written as (6) L n f(x) = x,j Pad n f(x,j )l,j (x), x = (x,x ), where l,j (x) are the fundamental interpolation polynomials uniquely determined by (7) l,j (x,j ) = δ, δ j,j, x,j Pad n, and l,j Π d n In the following section we derive an explicit formula for the fundamental polynomials

6 LEN BOS, STEFANO DE MARCHI, MARCO VIANELLO, AND YUAN XU 3 Construction of the Lagrange interpolation polynomials To derive explicits formulas for the fundamental Lagrange polynomials of (6), we will use orthogonal polynomials and follow the strategy in [8, 0] We note that the method in [8] wors for the case that V = dim Π n + σ with σ n In our case σ = n +, so that the general theory there does not apply We remind the reader that [3] gives an alternate derivation Let V d n denote the space of orthogonal polynomials of degree n with respect to () on [,] An orthonormal basis for V d n is given by P n(x,x ) := T n (x ) T (x ), 0 n, where T 0 (x) = and T (x) = T (x) for We introduce the notation P n = [P n 0,P n,,p n n ] and treat it both as a set and as a column vector The reproducing ernel of the space Π d n in L (W,[,] ) is defined by (3) K n (x,y) = n [P (x)] t P (y) = =0 n =0 j=0 Pj (x)pj (y) There is a Christoffel-Darboux formula (cf [6, 8]) which states that (3) K n (x,y) = [A n,ip n+ (x)] t P n (y) [A n,i P n+ (y)] t P n (x) x i y i, i =,, where x = (x,x ), y = (y,y ), and A n,i R (n+) (n+) are matrices defined by 0 0 0 0 0 A n, = 0 0, A n, = 0 0 0 0 0 0 0 The fundamental interpolation polynomials are given in terms of K n (x,y) To this end, we will try to express K n (x,y) in terms of the polynomials in the ideal I of Theorem 3 Recall that the Padua points are the common zeros of polynomials Q n+ defined in (4)and (5) We also denote [ (33) Q n+ := Q n+ 0,Q n+,,q n+ n, ] Q n+ n+ The definition of Q n+ shows that we have the relation (34) Q n+ = P n+ + Γ P n + Γ P n, where Γ R (n+) (n+) and Γ R (n+) n are matrices defined by 0 0 0 0 0 Γ = 0 and Γ = 0

BIVARIATE LAGRANGE INTERPOLATION AT PADUA POINTS 7 (35) By using the representation (34) we can rewrite (3) as where (x i y i )K n (x,y) =[Q n+ (x) Γ P n (x) Γ P n (x)] t A t n,ip n (y) P t n(x)a n,i [Q n+ (y) Γ P n (y) Γ P n (y)] =S,i (x,y) + S,i (x,y) + S 3,i (x,y), i =,, S,i (x,y) = Q t n+(x)a t n,ip n (y) P t n(x)a n,i Q n+ (y), S,i (x,y) = P t n(x) ( A n,i Γ Γ t A t n,i) Pn (y), S 3,i (x,y) = P t n(x)a n,i Γ P n (y) P t n (x)γ t A t n,ip n (y) If both x and y are in Pad n, then S,i will be zero We now wor out the other two terms First, observe that A n, Γ is a symmetrix matrix, 0 0 A n, Γ =, 0 0 and A n, Γ = It follows that S, (x,y) 0 and (36) S 3, (x,y) = P n 0 (x)p n 0 (y) + P n 0 (y)p n 0 (x) = T n (x )T n (y ) + T n (y )T n (x ) We also have A n, Γ = 0, which entails that S 3, (x,y) 0 Finally, we have A n, Γ Γ t A t n, =, from which we get (37) S, (x,y) = T n (x )T n (y ) T n (x )T n (y ) The terms S 3, and S, do not vanish on the Padua points We try to mae them part of the left hand side of (35) by seeing a polynomial h n (x,y) such that (38) It is easy to chec that S 3, (x,y) (x y )h n (x,y) = 0, x,y Pad n, S, (x,y) (x y )h n (x,y) = 0, x,y Pad n h n (x,y) := T n (x )T n (y )

8 LEN BOS, STEFANO DE MARCHI, MARCO VIANELLO, AND YUAN XU satisfies (38) This can be verified directly using the three term relation tt n (t) = T n+ (t) + T n (t) as follows By (36), S 3, (x,y) (x y )h n (x,y) = T n(x )(T n+ (y ) T n (y )) T n(y )(T n+ (x ) T n (x )) = T n(x )Q n+ 0 (y) T n(y )Q n+ 0 (x), using the definition of Q n+ 0 at (4) Similarly, by (37), S, (x,y) (x y )h n (x,y) = T n (y )(x T n (x ) + T n (x )) + T n (x )(y T n (y ) + T n (y )) = T n (y )Q n+ (x) + T n (x )Q n+ (y), using the definition of Q n+ at (5) Since Q n+ (x) vanishes on the Padua points, both equations in (38) are satisfied Consequently, we have proved the following proposition: Proposition 3 For x = (x,x ) and y = (y,y ), define (39) K n(x,y) := K n (x,y) T n (x )T n (y ) Then (x y )K n(x,y) = Q t n+(x)a t n,p n (y) P t n(x)a n, Q n+ (y) + T n(x )Q n+ 0 (y) T n(y )Q n+ 0 (x), (x y )K n(x,y) = Q t n+(x)a t n,p n (y) P t n(x)a n, Q n+ (y) T n (y )Q n+ (x) + T n (x )Q n+ (y) In particular, K (x,y) = 0 if x,y Pad n and x y Since P n n (x,x ) = T n (x ), we evidently have K n(x,x) > 0 Consequently, a compact formula for the fundamental interpolation polynomials follows immediately from the above proposition Theorem 3 The fundamental interpolation polynomials l,j in (6) associated with the Padua points x,j = (ξ,η j ) are given by (30) l,j (x) = K n(x,x,j ) K n(x,j,x,j ) = K n(x,x,j ) T n (x )T n (ξ ) K n (x,j,x,j ) [T n (ξ )] In fact, using the representation of K n in Proposition 3, the conditions (7) can be verified readily, which proves the theorem We can say more about the formula (30) In fact, a compact formula for the reproducing ernel K n appeared in [9], which states that (3) K n (x,y) =D n (θ + φ,θ + φ ) + D n (θ + φ,θ φ ) + D n (θ φ,θ + φ ) + D n (θ φ,θ φ ), where x = (cos θ,cos θ ), y = (cos φ,cos φ ), and D n (α,β) = cos((n + )α)cos α cos((n + )β)cos β cos α cos β

BIVARIATE LAGRANGE INTERPOLATION AT PADUA POINTS 9 Hence, the formula (30) provides a compact formula for the fundamental interpolation polynomials l,j Furthermore, the explicit formula (3) allows us to derive an explicit formula for the denominator K n(x,j,x,j ) The Padua points () naturally divide into three groups, Pad n = A v A b A in, where A v consists of the two vertex points (( ),( ) n ) for = 0,n, A b consists of the other points on the boundary of [,], and A in consists of interior points inside (,) (see Figure ) Proposition 33 For x,j Pad n, K, if x,j A v n(x,j,x,j ) = n(n + ), if x,j A b, if x,j A in The proof can be derived from the explicit formula of the Padua points () and the compact formula (3) by a tedious verification We refer to [3] for a proof using the generating curve Since integration of the Lagrange interpolation polynomial gives the quadrature formula, it follows from the above proposition and Corollary 4 that: Proposition 34 A cubature formula of degree n based on the Padua points is given by dx dx π f(x,x ) w [,] x x,j f(x,j ) x,j Pad n where w,j = /K n(x,j,x,j ) We note that this cubature formula is not a minimal cubature formula, since there are cubature formulas of the same degree with a fewer number of nodes 4 Convergence of the Lagrange interpolation polynomials In [3] it is shown that the Lagrange interpolation projection L n f in (6) has Lebesgue constant O((log n) ); that is, as an operator from C([,] ) to itself, its operator norm in the uniform norm is O((log n) ) This settles the problem of uniform convergence of L n f In this section we prove that L n f converges in L p norm For the Chebyshev weight function W defined in (), we define L p (W) as the space of Lebesgue measurable functions for which the norm ( ) /p f W,p := [,] f(x,x ) p W(x,x )dx dx is finite We eep this notation also for 0 < p <, even though it is no longer a norm for p in that range The main result in this section is Theorem 4 Let L n f be the Lagrange interpolation polynomial (6) based on the Padua points Let 0 < p < Then lim L nf f W,p = 0, f C([,] ) n

0 LEN BOS, STEFANO DE MARCHI, MARCO VIANELLO, AND YUAN XU In fact, let E n (f) be the error of the best approximation of f by polynomials from Π n in the uniform norm; then L n f f W,p c p E n (f) f C([,] ) The proof follows the approach in [0], where the mean convergence of another family of interpolation polynomials is proved We shall be brief whenever the same proof carries over First we need a lemma on the Fourier partial sum, S n f, defined by n S n f(x) = a j(f)pj (x) = π [,] K n f(x,y)w(y)dy =0 j=0 where a j (f) is the Fourier coefficient of f with respect to the orthonormal basis {Pj } in L (W) In the following we let c p denote a generic constant that depends on p only, its value may be different from line to line Lemma 4 Let < p < Then (4) S n f W,p c p f W,p, f L p (W) This is [0, Lemma 34] We will need another lemma whose proof follows almost verbatim from that of [0, Lemma 35] In the following we write N = Pad n = (n + )(n + )/ Lemma 43 Let p < Let x,j be the Padua points Then (4) P(x,j ) p c p P(x) p W(x)dx, P Π N n x,j Pad [,] p n The main tool in the proof of Theorem 4 is a converse inequality of (4), which we state below and give a complete proof, even though the proof is similar to that of [0, Theorem 33] Proposition 44 Let < p < Let x,j be the Padua points Then P(x) p W(x)dx c p P(x,j p, P Π d [,] N n x,j Pad n Proof Let P Π d n For p >, we have (43) P W,p = sup P(x)g(x)W(x)dx g W,q= [,] = sup P(x)S n g(x)w(x)dx, g W,q= [,] p + q =, where the second equality follows from the orthogonality We write S n g = S n g + a t n(g)p n, where a n = g(x)p n (x)w(x)dx [,]

BIVARIATE LAGRANGE INTERPOLATION AT PADUA POINTS Since the cubature formula is of degree n and PS n g is of degree n, we have P(x)S n (x)w(x)dx [,] = w,j P(x,j )S n (x,j ) x Pad,j n (44) /p /q x,j Pad n w,j P(x,j ) p By Lemma 4 and Lemma 43, x,j Pad n w,j S n (x,j ) q x,j Pad n w,j S n (x,j ) q c S n g W,q c g W,q = c Hence, using the fact that w,j N, it follows readily that (45) P(x)S n (x)w(x)dx [,] c w,j P(x,j ) p x,j Pad n We need to establish the similar inequality for the a t np n term Since L n P = P as P is of degree n, it follows from (30) and the orthogonality that P(x)a t n(g)p n (x)w(x)dx = L n P(x)a t n(g)p n (x)w(x)dx [,] [,] = w,j P(x,j ) K n(x,j,x)a t n(g)p n (x)w(x)dx x,j Pad [,] n = w,j P(x,j )a t n(g)p n (x,j ) an n(g) w,j P(x,j ) T n (ξ,j ) x,j Pad n x,j Pad n For the first term we can write a t (g)p n = S n g S n g and apply Hölder s inequality as in (44), so that the proof of (45) can be carried out again For the second term we use the fact that T n (x) to conclude that a n n(g) T n (x)g(x) W(x)dx g W,q [,] so that the sum is bounded by an n(g) w,j P(x,j ) T n (ξ,j ) x,j Pad n w,j P(x,j ) p x,j Pad n In this way we have established the inequality P(x)a t n(g)p n (x)w(x)dx [,] c w,j P(x,j ) p x,j Pad n Together with (45), this completes the proof of the proposition As shown in the proof of Theorem 3 of [0], the proof of Theorem 4, including the cases 0 < p <, follows as an easy consequence of Proposition 44 /p /p /p

LEN BOS, STEFANO DE MARCHI, MARCO VIANELLO, AND YUAN XU References [] L Bos, M Caliari, S De Marchi and M Vianello, A numerical study of the Xu polynomial interpolation formula in two variables Computing 76 (005), 3 34 [] L Bos, S De Marchi and M Vianello, On the Lebesgue constant for the Xu interpolation formula J Approx Theory 4 (006), 34 4 [3] L Bos, M Caliari, S De Marchi, M Vianello and Y Xu, Bivariate Lagrange interpolation at the Padua points: the generating curve approach J Approx Theory 43 (006), 5 5 [4] M Caliari, S De Marchi and M Vianello, Bivariate polynomial interpolation on the square at new nodal sets Appl Math Comput 65 (005), 6 74 [5] D Cox, J Little and D O Shea, Ideals, Varieties, and Algorithms, nd ed, Springer, Berlin, 997 [6] CF Dunl and Y Xu, Orthogonal Polynomials of Several Variables, Encyclopedia of Mathematics and its Applications, 8 Cambridge University Press, Cambridge, 00 [7] C R Morrow and T N L Patterson, Construction of algebraic cubature rules using polynomial ideal theory, SIAM J Numer Anal 5 (978), 953 976 [8] Y Xu, Common zeros of polynomials in several variables and higher dimensional quadrature, Pitman Research Notes in Mathematics Series, Longman, Essex, 994 [9] Y Xu, Christoffel functions and Fourier series for multivariate orthogonal polynomials J Approx Theory 8 (995), 05 39 [0] Y Xu, Lagrange interpolation on Chebyshev points of two variables J Approx Theory 87 (996), 0 38 [] Y Xu, Polynomial interpolation in several variables, cubature formulae, and ideals, Adv Comput Math (000), 363 376 Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada TNN4 E-mail address: lpbos@mathucalgaryca Department of Computer Science, University of Verona, 3734 Verona, Italy E-mail address: demarchi@sciunivrit Department of Pure and Applied Mathematics, University of Padua, 353 Padova, Italy E-mail address: marcov@mathunipdit Department of Mathematics, University of Oregon, Eugene, Oregon 97403-, USA E-mail address: yuan@mathuoregonedu