1.Introduction

Overview

The complexity of our algorithms will be measured by counting base field operations: we do not consider numerical issues (they may anyway be irrelevant, if e.g. our base field is a finite field), and do not discuss the choice of data structures or index manipulation issues.

From the complexity point of view, evaluation and interpolation are rather well understood for polynomials in one variable: algorithms of quasi-linear complexity are known to evaluate a polynomial of degree less than at points, and conversely to interpolate it. The best known algorithms [BM74] run in time , and the main remaining question is to close the gap between this and an optimal , if at all possible.

In several variables, the questions are substantially harder, due to the variety of monomial bases and evaluation sets one may consider; no quasi-linear time algorithm is known in general. In this paper, following the terminology of [Sau04], we consider evaluation points that are subgrids of tensor product grids. We prove that for some suitable monomial bases, evaluation and interpolation can both be done in time , where is the number of variables and is the size of the evaluation set (and of the monomial basis we consider). Remark that this result directly generalizes the univariate case. In many cases, is logarithmic in ; then, our result is optimal, up to logarithmic factors.

Moreover, for specific types of evaluation points, such as roots of unity or points in a geometric progression, even faster algorithms can be used in the univariate case, of time complexity . These algorithms will also be generalized to the multivariate case and result in evaluation and interpolation algorithms of time complexity , where is the maximal partial degree. In particular, given two dense multivariate polynomials in variables of total degree can be multiplied in time . To the best of our knowledge, this is the best currently available complexity bound for this problem. We also expect the new algorithms to be efficient in practice, although we have not implemented them yet.

Problem statement

In what follows, is a finite initial segment for the partial ordering on : this means that if and , then . For instance, one may think of as the set of standard monomials modulo a -dimensional ideal, for a given monomial ordering. Figure 1 shows such a set (black dots), as well as the minimal elements of (green squares).

As a very particular example, for positive integers , let denote the set ; this is an -dimensional grid.

The set will be used as an index set for both the evaluation points and the monomial basis. Let be our base field and let be such that . For , assume that we are given pairwise distinct elements ; we will denote by the collection . To we associate the point and we let : this will be our set of evaluation points. Remark that is contained in the “tensor product” grid

For instance, if for all , then and .

Let further be the polynomial ring in variables over ; for , we write . Then, denotes the -vector space of polynomials with support in . On the example of Figure 1, admits the monomial basis

Given a polynomial , written on the monomial basis, our problem of multidimensional multi-point evaluation is the computation of the vector .

Both the domain and the codomain of the evaluation map are -vector spaces of dimension , so it makes sense to ask whether this map is invertible. Indeed, let be the defining ideal of . A result going back to Macaulay (see [Mor03] for a proof) shows that the monomials form a monomial basis of . As a consequence, the former evaluation map is invertible; the inverse problem is an instance of multivariate interpolation.

Previous work

To obtain a quasi-linear result, we rely on the fast univariate algorithms of [BM74]. In the special case where is the grid , Pan [Pan94] solves the multivariate problem by applying a “tensored” form of the univariate algorithms, evaluating or interpolating one variable after the other. The key contribution of our paper is the use of a multivariate Newton basis, combined with fast change of basis algorithms between the Newton basis and the monomial basis; this will allow us to follow an approach similar to Pan's in our more general situation. The Newton basis was already used in many previous works on our interpolation problem [Wer80, Müh88], accompanied by divided differences computations: we avoid divided differences, as they lead to quadratic time algorithms.

The results in this paper have a direct application to multivariate power series multiplication. Let be as above, and let be the monomial ideal generated by ; equivalently, is generated by all minimal elements of . Then, one is interested the complexity of multiplication modulo , that is, in . Suitable choices of lead to total degree truncation (take , so ), which is used in many forms of Newton-Hensel lifting, or partial degree truncation (take , so ).

There is no known algorithm with quasi-linear cost for this question in general. Inspired by the sparse multiplication algorithm of [CKL89], Lecerf and Schost gave such an algorithm for total degree truncation [LS03]. It was extended to weighted total degree in [vdH02] and further improved from the bit-complexity point of view in [vdHL09]. Further speed-ups are possible in small dimensions, when using the Truncated Fourier Transform or TFT [vdH04, vdH05]. For more general truncation patterns, Schost [Sch05] introduced an algorithm based on deformation techniques that uses evaluation and interpolation of the form described in this paper. At the time of writing [Sch05], no efficient algorithm was known for evaluation and interpolation; the present paper fills this gap and completes the results of [Sch05].

Conventions

In all that follows, we let denote a multiplication time function, in the sense that univariate polynomials of degree less than can be multiplied in operations in . As in [GG03], we impose the condition that is an increasing function (and freely use all consequences of this assumption), and we note that can be taken in using the algorithm of [CK91].

We will use big-Oh notation for expressions that depend on an unbounded number of variables (e.g., for Lemma 6 below). In such cases, the notation means that there exists a universal constant such that for all and all , the inequality holds.

2.Univariate Algorithms

This section describes some mostly classical algorithms for univariate polynomials over . We denote by the set of univariate polynomials of degree less than . Given pairwise distinct points in , we write for . The polynomials are called the Newton basis associated to ; they form a -basis of . For instance, for , we have .

Because we will have to switch frequently between the monomial and the Newton bases, it will be convenient to use the notation , for , with

We write to indicate that a polynomial is written on the basis ; remember that when no value is mentioned in subscript, we are working in the Newton basis.

2.1.General results

The following classical lemma [BP94, Ex. 15 p. 67] gives complexity estimates for conversion between these bases.

Evaluation and interpolation in the monomial basis can be done in time , by the algorithms of [BM74]; combining this to the previous lemma, we obtain a similar estimate for evaluation and interpolation with respect to the Newton basis.

If the points are in geometric progression, we may remove a factor in all estimates. Indeed, under these assumptions, the conversions of Lemma 1 and the evaluation or interpolation of Lemma 2 take time [BS05]. The following subsection studies in detail another particular case, TFT (Truncated Fourier Transform) points. In this case we may also remove the factor in all estimates, but the constant factor is even better than for points in geometric progression.

2.2.TFT points

In this subsection, we are going to assume that contains suitable roots of unity, and prove refined complexity bounds for such points.

Let be as above, let be the smallest integer such that and let us suppose that contains a primitive -th root of unity . The TFT points are , for , where is the binary -bits mirror of . In other words, they form an initial segment of length for the sequence of -th roots of unity written in the bit-reverse order; when for instance and , these points are . In this subsection, these points are fixed, so we drop the subscript in our notation.

It is known [vdH04, vdH05] that in the monomial basis, evaluation and interpolation at the TFT points can be done in time . Precisely, both operations can be done using shifted additions and subtractions and multiplications by powers of . Here we recall that a shifted addition (resp. subtraction) is a classical addition (resp. subtraction) where any of the inputs may be premultiplied by or (e.g., ). In the most interesting case , the TFT roughly saves a factor of 2 over the classical FFT; this makes it a very useful tool for e.g. polynomial multiplication.

We will show here that similar results hold for the conversion between the monomial and Newton bases. First, we give the basis of the algorithm by assuming that , so that . Such a polynomial can be written in the monomial, resp. Newton basis, as

For , let us introduce the polynomials , all of degree less than , such that

(1)

Thus, for and , whereas for .

Lemma 3. For and , we have

Proof. This follows by grouping the terms of indices and in (1), and by noticing that for all , the following equality holds:

Indeed, for all even , ; thus, the left-hand side is the product of all , for . The claim follows by writing , observing that .

The previous lemma implies an algorithm of complexity that takes as input the coefficients on the Newton basis, and outputs on the monomial basis. It suffices to compute all polynomials (on the monomial basis) by means of the recursive formula. Computing from and takes additions and multiplications by powers of , so going from index to takes a total of additions and multiplications. The inverse conversion takes time as well, since knowing , we can recover first for free as the high-degree terms of , then using additions and multiplications.

The conversion algorithm from the Newton to the monomial basis can be depicted as follows, in the case , . The flow of the algorithm goes down, from to ; the th row contains the 16 coefficients of the polynomials (on the monomial basis), in that order, and each oblique line corresponds to a multiplication by a root of unity. The algorithm does only “half-butterflies”, compared to the FFT algorithm.

Figure 2. Schematic representation of the conversion Newton to monomial for

If we assume that , we will be able to avoid useless computations, by keeping track of the zero coefficients in the polynomials . The next figure shows the situation for ; this is similar to what happens in van der Hoeven's TFT algorithm, but much simpler (here, at each level, we can easily locate the zero coefficients).

Figure 3. Schematic representation of the conversion Newton to monomial for

The pseudo-codes for the conversion from Newton basis to monomial basis and the inverse transformation are as follows:

We deduce the following complexity result for the conversions, which refines Lemma 1; it is of the form , with a tight control on the constants.

Proof. For any given , we do additions/subtractions and multiplications. This is at most .

The equivalent of Lemma 2 for the TFT points comes by using van der Hoeven's TFT algorithms for evaluation and interpolation on the monomial basis, instead of the general algorithms.

3.Projections and Sections

Let be a finite initial segment and let be such that is contained in . We present here some geometric operations on that will be useful for the evaluation and interpolation algorithms.

Projections

We will denote by the projection

of on the -coordinate plane. For in , we let be the unique integer such that and . In particular, holds for all .

In Figure 1, we have ; consists of the points of ordinates on the vertical axis, with , , and .

Finally, if is a collection of points as defined in the introduction, with for all , then we will write .

Sections

For , we let be the section

and we let be the projection of on the -coordinate plane. In other words, is in if and only if is in . We have the following equivalent definition

Because the sets form a partition of , we deduce the equality . Notice also that all are initial segments in . In Figure 1, we have , , , and .

4.Multivariate Bases

From now on, we focus on multivariate polynomials. In all this section, we fix a finite initial segment and such that . Naturally, polynomials in may be written in the monomial basis , but we may also use the multivariate Newton basis , defined by

Generalizing the univariate notation, given , we will consider a mixed monomial-Newton basis with

As in the univariate case, we will write to indicate that is written on the corresponding basis.

It will be useful to rely on the following decomposition. Let be in , written on the basis . Collecting coefficients, we obtain

(2)

with and

(3)

Keep in mind that if the indices and are omitted, we are using the Newton basis.

Lemma 5. Let be in , and let be obtained by replacing by in , for some in . Let be in . Given , one can compute in time

Proof. Using a permutation of coordinates, we reduce to the case when . Using the above notations, it suffices to convert from the basis to the basis for all . By Lemma 1, each conversion can be done in time , so the total cost is

Since the function is increasing, we get the upper bound

the conclusion follows from the equality .

Let us write and , where both vectors have length . Then, the basis is the monomial basis, whereas the basis is the Newton basis. Changing one coordinate at a time, we obtain the following corollary, which shows how to convert from the monomial basis to the Newton basis, and back.

Lemma 6. Let be in and let be in . Given , one can compute in time

The remarks in Section 2 about special families of points apply here as well: if the points in have special properties (e.g., is in geometric progression, or the are TFT points), the cost may be reduced (both in the geometric case and in the TFT case, we may save the factors ).

5.Multivariate Evaluation and Interpolation

We are now in a position to state and prove our main result.

Theorem 7. Given such that , with written on the monomial basis of , one can evaluate at in time

Conversely, given the values of at , one can compute the representation of on the monomial basis of with the same cost.

Using the bound , and the fact that holds for all , we deduce the simplified bound claimed in the introduction. Remark also that our result matches the cost of the algorithm of [Pan94], which applies in the special case of evaluation-interpolation at a grid.

The input to the evaluation algorithm is given on the monomial basis of ; however, internally to the algorithm, we use the Newton basis. Thus, before entering the (recursive) evaluation algorithm, we switch once and for all to the Newton basis; this does not harm complexity, in view of Lemma 6. Similarly, the interpolation algorithm uses the Newton basis, so we convert the result to the monomial basis after we have completed the interpolation.

Remark also that if the points are in geometric progression for each , then one may eliminate the factors from the complexity bound. Using TFT evaluation points allows for similar reductions.

5.1.Setup

The algorithm follows a pattern similar to Pan's multivariate evaluation and interpolation at a grid [Pan94]: e.g. for evaluation, we evaluate at the fibers above each , and proceed recursively with polynomials obtained from the sections . Using the Newton basis allows us to alleviate the issues coming from the fact that is not a grid.

Let be written (in the Newton basis) as in the previous section:

where we write and

To , we associate the -variate polynomial

The key to our algorithms is the following proposition.

Proposition 8. For all , the following equality holds:

Proof. First, we make both quantities explicit. The left-hand side is given by

whereas the right-hand side is

where in both cases we write . Thus, to conclude, it is enough to prove that, for , we have .

Indeed, recall that the assumption implies . On the other hand, we have , whence the inequality , where we write . In particular, we deduce , which in turn implies that . Thus, there exists such that . This implies that , as requested.

5.2.Evaluation

Given , with written in the Newton basis , we show here how evaluate at . The algorithm is the following.

• If , is a constant; we return it unchanged.

• Otherwise, we compute all values , for and , by applying the fast univariate evaluation algorithm to each . For , and for , we have (by definition) , so we have all the information we need to form the polynomial . Then, we evaluate recursively each at , for .

Proposition 9. The above algorithm correctly evaluates at in time

Proof. Correctness follows directly from Proposition 8, so we can focus on the cost analysis. Let denote the cost of this algorithm. The former discussion shows that and that is the sum of two contributions:

• the cost of computing all values , for and

• the cost of the recursive calls on for .

Lemma 2 shows that the former admits the upper bound

as in the proof of Lemma 5, this can be bounded by

As to the recursive calls, notice that all are contained in , which is contained in . Thus, for some constant , we obtain the inequality

(4)

To conclude, we prove that for all , for all and for any initial segment , we have

(5)

Such an inequality clearly holds for . Assume by induction on that for any and any initial segment , we have

(6)

To prove (5), we substitute (6) in (4), to get

Since , we are done.

5.3.Interpolation

The interpolation algorithm is obtained by reversing step-by-step the evaluation algorithm. On input, we take , with ; the output is the unique polynomial such that for all .

• If , consists of a single entry; we return it unchanged.

• Otherwise, we recover recursively all , for . This is made possible by Proposition 8, which shows that we actually know the values of each at the corresponding . Knowing all gives us the values for all and . It suffices to interpolate each on the Newton basis to conclude.

Correctness of this algorithm is clear and the following complexity bound is proved in a similar way as in the case of evaluation.

Proposition 10. The above algorithm correctly computes in time

6.Applications

We conclude with an application of our results to the multiplication of polynomials and power series. Let and be as above. We let , as we assume that has cardinality at least , so that we can find , where consists of pairwise distinct entries in . Let , so that .

6.1.Multiplication of polynomials

We discuss here the case when we want to multiply two polynomials and with . In this case, we may use a simple evaluation-interpolation strategy.

• Perform multi-point evaluations of and at ;

• Compute the componentwise product of the evaluations;

• Interpolate the result at to yield the product .

By Theorem 7, this can be done in time

If admits at least points in geometric progression (or at least TFT points), then the factor may be removed. This result should be compared to the algorithm of [CKL89], which has complexity ; that algorithm applies to more general monomial supports, but under more restrictive conditions on the base field.

6.2.Multiplication of power series

Let now be the monomial ideal generated by . We discuss here the complexity of multiplication modulo in . To our knowledge, no general algorithm with a complexity quasi-linear in is known.

Let us first recall an algorithm of [Sch05] and show how our results enable us to improve it. Theorem 1 of [Sch05] gives an algorithm for multiplication in , that relies on the following operations:

• multi-point evaluations at of polynomials in ;

• univariate power series multiplication in precision ;

• interpolations at of polynomials in .

The paper [Sch05] does not specify how to do the evaluation and interpolation (for lack of an efficient solution); using our results, it becomes possible to fill all the gaps in this algorithm. Applying Theorem 7, without doing any simplification, we obtain a cost of

Using the inequality , this gives the upper bound . If we can take at least points in geometric progression in (or at least TFT points), then the upper bound reduces to .

6.3.Power series with total degree truncation

The most important case of truncated power series multiplication is when we truncate with respect to the total degree. In other words, we take . In that case, several alternative strategies to the one of the former subsection are available [LS03, vdH04, vdH05, vdHL09], and we refer to [vdH05, vdHL09] for some benchmarks.

As it turns out, one can apply the result from Section 6.1, in the special case of polynomials supported in total degree, to improve these algorithms, when admits at least points in geometric progression (or at least TFT points). Indeed, the algorithms from [LS03, vdHL09] rely on multivariate polynomial multiplication. Using the result of Section 6.1 in these algorithms (instead of sparse polynomial multiplication), we obtain a new algorithm of time complexity instead of . For constant , this removes a factor from the asymptotic time complexity.

To finish, we would like to point out that the present paper almost repairs an error in [vdH04, Section 5], which was first announced in [vdH05]. Indeed, it was implicitly, but mistakenly, assumed that Proposition 8 also holds for monomial bases. The present “fix” simply consists of converting to the Newton basis before evaluating, and similarly for the inverse. The asymptotic time complexity analysis from [vdH04, Section 5] actually remains valid up to a small but non trivial constant factor. When using TFT transforms in combination with the algorithms from section 2.2, we expect the constant factor to be comprised between one and three in practice.

Bibliography