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# Waring’s problem for polynomial rings and the digit sum of exponents

- Seth Dutter
^{1}Email authorView ORCID ID profile and - Cole Love
^{1}

**Received:**7 September 2016**Accepted:**23 November 2016**Published:**1 February 2017

## Abstract

Let \(\overline{\mathbb {F}}_p\) be the algebraic closure of the finite field \(\mathbb {F}_p\). In this paper we develop methods to represent arbitrary elements of \(\overline{\mathbb {F}}_p[t]\) as sums of perfect *k*-th powers for any \(k\in \mathbb {N}\) relatively prime to *p*. Using these methods we establish bounds on the necessary number of *k*-th powers in terms of the sum of the digits of *k* in its base-*p* expansion. As one particular application we prove that for any fixed prime \(p > 2\) and any \(\epsilon > 0\) the number of \((p^r-1)\)-th powers required is \(\mathcal {O}\left( r^{(2+\epsilon )\ln (p)}\right) \) as a function of *r*.

## Keywords

- Waring’s problem
- Function fields
- Positive characteristic

## Mathematics Subject Classification

- 11P05
- 11T55

## 1 Background

Waring’s problem asks whether there exists some positive integer *s* such that every natural number can be represented as the sum of at most *s*
*k*-th powers. Vaughan and Wooley [9] provide a comprehensive survey of the classic version of this problem. Rather than focus on natural numbers, we consider the problem of representing an arbitrary polynomial as a sum of *k*-th powers over a field of positive characteristic. This question and its variants have been extensively studied. Paley [6] was the first to establish upper bounds on the necessary size of *s* in the case of polynomials over finite fields. More recently, Vaserstein [8] and Liu and Wooley [3] have improved upon these bounds. In the characteristic 0 case Newman and Slater [5] have proven lower and upper bounds on the number of *k*-th powers needed to represent an arbitrary polynomial.

The characteristic of the field plays a crucial role in determining not only the number of *k*-th powers necessary, but whether the problem even has a solution. For example, over a field of characteristic 2 it is never possible to write the monomial *t* as a sum of perfect squares due to the Frobenius endomorphism. On the other hand, the upper bounds established in [8] for the positive characteristic case are often lower than the proven lower bound in [5] for the characteristic 0 case. Even with a fixed positive characteristic, the size of the field plays an important role, as algebraic extensions may be necessary for certain constructions.

*strict*if \(\deg (y_i^k) < \deg (f)+k\) for all \(i\in \{1,\ldots , s\}\). See [4] for a thorough treatment of the strict case. Here we will investigate the

*unrestricted*variant of Waring’s problem wherein no condition is placed upon the degree of the \(y_i\). A consequence of studying the unrestricted variant is that it suffices to represent

*t*as a sum of

*k*-th powers. Indeed, if

*f*(

*t*). As we are only handling the unrestricted variant in this paper, the proofs will focus on representing

*t*as a sum of

*k*-th powers.

Throughout this paper we will let \(\overline{\mathbb {F}}_p\) denote the algebraic closure of \(\mathbb {F}_p\) where *p* is prime. If *q* is a power of *p* we will naturally identify \(\mathbb {F}_q\) with the subfield of \(\overline{\mathbb {F}}_p\) having *q* elements. As in [10] we let \(\upsilon (p,k)\) denote the smallest natural number *s* such that every polynomial in \(\overline{\mathbb {F}}_p[t]\) can be written as a sum of at most *s*
*k*-th powers. If no such *s* exists, then we say \(\upsilon (p,k) = \infty \).

*q*expansion of

*k*. When doing so we will write

*k*will be assumed relatively prime to

*q*and therefore \(a_1 = 0\). The

*k*studied in this paper will usually be sparse in the sense that most digits are 0 and for this reason we will only list the nonzero \(k_i\) in the expansion.

*p*expansion of

*k*, then Theorem 2 of Vaserstein [8] establishes the bound

*k*relatively prime to

*p*. Our results are directed at improving this bound, under certain conditions, by relating \(\upsilon (p, k)\) to the sum of the digits in the base-

*q*expansion of

*k*for some

*q*. As such, we introduce the notation

*k*-th powers. Thus, it suffices to represent

*t*as a linear combination of

*k*-th powers of polynomials.

We now state our main theorem, the proof of which is provided in Sect. 2.

### Theorem 1

*p*be prime and \(k\in \mathbb {N}\) be relatively prime to

*p*. If

*M*, \(n\in \mathbb {N}\) satisfy the conditions \(\gamma _{p^n}(k)\le M\) and \(M-1\mid p^n-1\), then

### Remark 1

If \(k \mid p^r+1\) for some natural number *r*, then we can take \(M=2\) and \(n=1\) in Theorem 1 to get \(\upsilon (p, k) \le 3\). This particular case also follows from Eq. 1 and has been additionally studied by Car [1, 2] and Voloch [10].

It should be noted that while we are working over an algebraically closed field all constructions will take place over some finite subfield of effectively computable size. In most instances the size of this subfield will be much larger than those needed for the constructions in [1, 2, 8, 10]. However, when applicable in base-*p*, the bound established in Theorem 1 frequently gives an improvement over the bound in Eq. 1. While only a small proportion of numbers meet the hypothesis in base-*p*, in practice this limitation can often be worked around by using a larger base. In Sect. 3 we give examples of how this can be done, the results of which now follow.

### Theorem 2

*p*be an odd prime, \(k \in \mathbb {N}\) be relatively prime to

*p*, and

*p*have odd order in \((\mathbb {Z}/k\mathbb {Z})^\times \). If

*c*is the least nonnegative integer such that \(\gamma _p(k) \le 2^c+1\), then

### Remark 2

If *p* and \(\ell \) are distinct primes with *p* odd, then either *p* has odd order in \((\mathbb {Z}/\ell \mathbb {Z})^\times \) or there exists some natural number *r* such that \(p^r\equiv -1\pmod {\ell }\). In the latter case, by Remark 1, we have \(\upsilon (p,\ell ) \le 3\). In the former case, by Theorem 2, we have \(\upsilon (p, \ell ) < 4\gamma _p(\ell )\). In either case, the bound is small when the exponent is prime.

In the event that *p* does not have odd order, or that \(p=2\), bounds still follow from Theorem 1, but they are no longer linear in \(\gamma _p(k)\).

### Theorem 3

*p*be prime, \(k\in \mathbb {N}\) be relatively prime to

*p*, and

*r*the order of

*p*in \((\mathbb {Z}/k\mathbb {Z})^\times \). If \(u\in \mathbb {N}\) is relatively prime to

*r*and \(\gamma _p(k) \le p^u\), then

### Remark 3

*c*is the least nonnegative integer such that \(\gamma _2(k) \le 2^{2^c}\), then, in the notation of Theorem 3, letting \(u = 2^c\) gives the bound

A consequence of the bound in Eq. 1 is that \(\upsilon (p, k) < k\) for all \(k\in \mathbb {N}\) satisfying the following conditions: *k* is relatively prime to *p*, \(k > p\), and *k* is not of the form \(p^r-1\) for any \(r\in \mathbb {N}\). In the case that \(k=p^r-1\) the established bound is \(\upsilon (p, p^r-1) \le p^r-1\). If *p* and *r* are odd, Theorem 2 gives the generally sharper bound of \(\upsilon (p, p^r-1) < 4(p-1)r\). However, this theorem does not apply if either *p* or *r* are even. In the following theorem we establish that \(v(p,p^r-1)\) is bounded by a polynomial in *r* of degree determined by *p*. The proof of this result appears in Sect. 3 and relies on bounds for the first Chebyshev function.

### Theorem 4

*p*be prime and \(\epsilon > 0\) be a real number. Then

*r*tends to infinity, where the constants involved in the big \(\mathcal {O}\) notation depend upon \(\epsilon \) and

*p*.

## 2 The main theorem

The structure of the proof of Theorem 1 is as follows. We begin with Lemma 1 by proving the existence of some \(k'\) having properties which are necessary for subsequent constructions and such that \(k | k'\). Without a loss of generality we will therefore assume that *k* meets the conditions of the conclusion of Lemma 1. In Lemma 2 we establish a multivariate polynomial identity over \(\overline{\mathbb {F}}_p\) which, after a suitable sequence of substitutions, gives a univariate identity with a degree 1 polynomial, some terms of degree *k*, and some terms of intermediate degrees. The properties assumed on *k* allow for a linear combination of two of these identities to eliminate all terms of intermediate degrees.

### Lemma 1

*q*and

*k*be relatively prime natural numbers and

*q*expansion of

*k*. Then there exists \(k^\prime \), \(u \in \mathbb {N}\) satisfying the following conditions: \(u > 1\), \(k | k^\prime \), and the base-

*q*expansion of \(k^\prime \) is given by

### Proof

Denote by *r* the order of *q* in \((\mathbb {Z}/k\mathbb {Z})^\times \) and let \(u > 1\) be relatively prime to *r*. For each \(i\in \{2,\ldots ,N\}\) there are infinitely many integers \(b_i\) satisfying \({r b_i + a_i \equiv 1 \pmod {u}}\). Select the \(b_i\) such that the sequence \(a_i^\prime = r b_i + a_i\) is positive and strictly increasing. Let \(k^\prime \) be given as in the statement of the lemma in terms of these \(a_i'\). Since \(q^r \equiv 1 \pmod {k}\) we have \(q^{a_i^\prime } \equiv q^{rb_i}q^{a_i} \equiv q^{a_i}\pmod {k}\). From which it follows that \(k^\prime \equiv k\pmod {k}\) and therefore \(k|k^\prime \). \(\square \)

### Remark 4

Note that by construction we have \(\gamma _q(k) = \gamma _q(k^\prime )\). In particular, \(k^\prime \) can be used in place of *k* without changing the hypothesis or bound of Theorem 1.

The following lemma makes no assumption on the characteristic of the underlying field. Although in general we assume that the field is algebraically closed and of positive characteristic, Lemma 2 only requires the existence of a suitable number of roots of unity.

### Lemma 2

*F*be a field and \(\omega \in F\) be a primitive \((M-1)\)-th root of unity for some \(M\ge 2\). Let \(k_1,\ldots ,k_N\) be natural numbers such that \(\sum _{i=1}^N k_{i} = M\). Then the following identity holds in the polynomial ring \(F[x_1,\ldots , x_N]\).

### Proof

*Case 1*Suppose that \(D = M\). Then \(d_i = k_i\) for all

*i*. The monomial is \(\prod _{i=1}^N x_i^{k_i}\) and the coefficient is

*Case 2*Suppose that \(D=1\). Then exactly one \(d_i\) is equal to 1 and the rest are 0. In which case, the monomial is \(x_i\) and the coefficient is

*Case 3*Suppose that \(D \notin \{1, M\}\). If \(M = 2\), then \(D=0\) and \(\omega = 1\). This corresponds to the constant term in the expansion of

Combining these three cases gives the desired identity. \(\square \)

We will assume without further mention that *k* meets the conditions of the conclusion of Lemma 2. See Remark 4 for details.

### Proof of Theorem 1

*q*expansion of

*k*. The first step is to replace all of the multivariate products in Equation 3 with univariate polynomials raised to the

*k*-th power. Since \(M-1 | q-1\) and \(q-1 | q^{a_i}-1\), it follows that \((\omega ^j)^{q^{a_i}-1} = 1\), so \((\omega ^j)^{q^{a_i}} = \omega ^j\) for all integers

*j*and all \(i\in \{2, \ldots , N\}\). Therefore, working in \(\overline{\mathbb {F}}_p[t]\), we have

*q*expansion of

*k*. After applying Eq. 4 and separating out the linear term, we get

*u*is such that \(a_i \equiv 1 \pmod {u}\) for \(i\in \{2,\ldots ,N\}\). Let \(\lambda \) be a generator of the group \(\mathbb {F}_{q^u}^\times \subset \overline{\mathbb {F}}_p\). For each \(i\in \{2,\ldots , N\}\) write \(a_i = g_i u + 1\) for some nonnegative integer \(g_i\). Observe that \(q^{a_i}-q = q(q^{g_i u}-1)\) from which it follows \(q^u-1 \mid q^{a_i}-q\). Since \(\lambda \) has order \(q^u-1\) we have \(\lambda ^{q^{a_i}-q} = 1\) and therefore \(\lambda ^{q^{a_i}} = \lambda ^q\). In order to obtain another equation to work with we replace

*t*with \(\lambda t\) in Eq. 5 to get

*k*is relatively prime to

*p*, \(k_1 \ne 0\), so the coefficient of

*t*is nonzero. After rearranging terms we get a degree one polynomial in

*t*equal to a sum of \(2(M-1)+1\) perfect

*k*-th powers. Therefore, \(v(p,k) \le 2(M-1)+1\) when \(\gamma _q(k) = M\).

*k*-th power in the final equation, so \(\upsilon (p,k) \le 2(M-1)\) when \(\gamma _q(k) < M\). \(\square \)

It should be noted that Eq. 5 can also be established as a consequence of Lucas’ theorem. To do so requires an analysis of the base-*q* digits of numbers congruent to 1 modulo \(M-1\) and the breakdown of these digits into their base-*p* expansion. The authors have instead opted for the proof given above as it is self-contained and provides their original insight into the problem.

## 3 Applications of the main theorem

The primary weakness of Theorem 1 is the requirement that the sum of the digits be at most the base we are working over. The following lemma allows us to systematically increase the base to get some \(k'\), divisible by *k*, without changing the sum of the digits. This pumping of *k* is the main tool used to prove the remaining theorems.

### Lemma 3

Let *q*, \(k \in \mathbb {N}\) be relatively prime and *q* have order *r* in \((\mathbb {Z}/k\mathbb {Z})^\times \). For any \(u\in \mathbb {N}\), relatively prime to *r*, there exists some natural number \(k^\prime \) such that \(k\mid k^\prime \) and \(\gamma _{q^u}(k^\prime ) = \gamma _q(k)\).

### Proof

*u*and

*r*are relatively prime there exists some \(b\in \mathbb {N}\) such that \(rb\equiv -1 \pmod {u}\). By our hypothesis, \(q^r \equiv 1\pmod {k}\), it follows that \(q^{rb+1}\equiv q\pmod {k}\). Let \(k_1+k_2q^{a_2}+\cdots +k_N q^{a_N}\) be the base-

*q*expansion of

*k*and define \(k'\) by replacing

*q*with \(q^{rb+1}\) in this expansion. Then we have

*q*expansion of

*k*. Therefore, \(\gamma _{q^u}(k^\prime ) = \gamma _q(k)\). \(\square \)

With Lemma 3 in place we now proceed to prove Theorem 2 and Theorem 3.

### Proof of Theorem 2

*r*be the order of

*p*in \((\mathbb {Z}/k\mathbb {Z})^\times \) and

*c*be the least nonnegative integer such that \(\gamma _p(k) \le 2^c+1\). In the notation of Lemma 3 we let \(u = 2^c\), in which case there exists some \(k^\prime \), divisible by

*k*, satisfying \(\gamma _{p^u}(k') = \gamma _{p}(k)\). Since

*p*is odd it has order dividing \(2^c\) in \((\mathbb {Z}/2^{c+1}\mathbb {Z})^\times \), and thus \(2^{c+1} \mid p^u-1\), so \(2^c\mid p^u-1\). Let \(M = 2^c+1\) and \(n = u\) in Theorem 1 to get \(v(p, k') \le 2^{c+1}+\mathbf {1}_{2^c+1}(\gamma _{p^u}(k'))\). Since \(\upsilon (p, k^\prime ) \le \upsilon (p, k^\prime )\) and \(\gamma _{p^u}(k^\prime ) = \gamma _{p}(k)\) we have

### Proof of Theorem 3

Let *u* be as in the statement of the theorem. By Lemma 3 there exists some \(k^\prime \), divisible by *k*, such that \(\gamma _{p^u}(k^\prime ) = \gamma _p(k)\le p^u\). Let \(M = p^u\) and \(n=u\) in Theorem 1 to get \(\upsilon (p, k^\prime ) \le 2(p^u-1)+\mathbf {1}_{p^u}(\gamma _{p^u}(k'))\). Since \(\upsilon (p, k) \le \upsilon (p, k^\prime )\) and \(\gamma _{p^u}(k^\prime ) = \gamma _{p}(k)\) the result immediately follows. \(\square \)

*x*define

*r*.

### Proof of Theorem 4

*N*, depending on \(\epsilon \), such that for all \(x > N\) the inequality

*r*is large enough, so that \(\ln (r) > N\), then

*r*cannot be divisible by all primes between \(\ln (r)\) and \((2+\epsilon )\ln (r)\). Let \(\ell \) be a prime in this range such that \(\ell \not \mid r\). By construction \(p^\ell \ge p^{\ln (r)} = r^{\ln (p)}\). Since \(\ln (p) > 1\), for

*r*sufficiently large we have \(r^{\ln (p)} > r(p-1) = \gamma _p(p^r-1)\). Apply Lemma 3 with \(u = \ell \) to get some \(k^\prime \) satisfying \(p^r-1 | k^\prime \) and \(\gamma _{p^\ell }(k^\prime ) = r(p-1)\). Consequently, we have \(\gamma _{p^\ell }(k^\prime ) \le p^\ell \). In the notation of Theorem 1, let \(M = p^\ell \) and \(n=\ell \) to get

*r*.

*r*. \(\square \)

## Declarations

### Open Access

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

### Acknowledgements

Research supported in part by the UW–Stout Foundation.

### Competing interests

The authors declare that they have no competing interests.

## Authors’ Affiliations

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