    # On the Degree of Boolean Functions as Polynomials over Z_m

Polynomial representations of Boolean functions over various rings such as Z and Z_m have been studied since Minsky and Papert (1969). From then on, they have been employed in a large variety of fields including communication complexity, circuit complexity, learning theory, coding theory and so on. For any integer m>2, each Boolean function has a unique multilinear polynomial representation over ring Z_m. The degree of such polynomial is called modulo-m degree, denoted as deg_m(·). In this paper, we discuss the lower bound of modulo-m degree of Boolean functions. When m=p^k (k> 1) for some prime p, we give a tight lower bound that deg_m(f)≥ k(p-1) for any non-degenerated function f:{0,1}^n→{0,1}, provided that n is sufficient large. When m contains two different prime factors p and q, we give a nearly optimal lower bound for any symmetric function f:{0,1}^n→{0,1} that deg_m(f) ≥n/2+1/p-1+1/q-1. The idea of the proof is as follows. First we investigate properties of polynomial representation of MOD function, then use it to span symmetric Boolean functions to prove the lower bound for symmetric functions, when m is a prime power. Afterwards, Ramsey Theory is applied in order to extend the bound from symmetric functions to non-degenerated ones. Finally, by showing that deg_p(f) and deg_q(f) cannot be small simultaneously, the lower bound for symmetric functions can be obtained when m is a composite but not prime power.

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## 1 Introduction

Given a Boolean function , the degree (resp., modulo- degree), denoted as (resp., ), is the degree of the unique111The existence and uniqueness are guaranteed by the Möbius inversion, see e.g. . multilinear polynomial representation over (resp., ). These complexity measures and related notions have been well studied since the work of Minsky and Papert . The polynomial representation of a Boolean function has found numerous applications in the study of query complexity (see e.g. ), communication complexity [4, 26, 29, 28, 27, 22, 9], learning theory [16, 19, 15, 23], explicit combinatorial constructions [12, 13, 10, 6], circuit lower bounds [31, 25, 1, 11] and coding theory [33, 34, 14, 20].

In this paper, we focus on modulo- degree of Boolean functions. One of the complexity theoretic motivations of studying is to understand the power of modular counting. The famous Razborov–Smolensky polynomial method [25, 31] reduces the task of proving size lower bounds for circuits to proving the lower bound of approximate modulo- degree of the target Boolean function. However, their idea only works when is a prime.222It is a folklore that , where is the square-free part of . Therefore in fact we are able to handle circuits for any prime power . Nevertheless, it is still important to understand the computational power of polynomials over for general .

Towards the complexity measure itself, the case when is a prime has been studied a lot in previous works. For example, one natural question is whether is polynomially related to for general

, as other complexity measures like decision tree complexity

do? The answer is a NO according to the parity function . That is, but . Though this function works as a counterexample for the relationship between and , it is still inspiring because its modulo- degree seems to be large. After some careful calculation, one can get . Actually, Gopalan et al.  give the following relationship between the polynomial degrees modulo two different primes and :

 degq(f)≥n⌈log2p⌉degp(f)p2degp(f).

Daunting at the first glance, the inequality implies an essential fact that, as long as , a lower bound of for follows. Moreover, if has at least two different prime factors and , then .

Having negated the possibility for the case of prime , it is natural to study the case of composite modulo. The systematic study of this case was initiated by Barrington et al. . Alas, whether is polynomially related to is still a widely open problem. Though the answer for the case being prime power is proved to be a NO in Gopalan’s thesis , we are unable to find better separation between and , for with and being two distinct primes, than the quadratic one given by Li and Sun . This leads to the following conjecture:

###### Conjecture 1.

Let be a Boolean function. If has at least two distinct prime factors, then,

 deg(f)=O(poly(degm(f))).

Towards this conjecture, the first step is to deal with symmetric Boolean functions. Lee et al.  proves that for any distinct primes and non-trivial symmetric Boolean function , implying the correctness of Conjecture 1 in symmetic cases. Li and Sun  improved their bound to , which implies . This is far from being tight; actually, as we will present later, and can be eliminated from the denominator.

On the tight lower bound of , Nisan and Szegedy  give the bound as long as is non-degenerated. Note that this bound is tight up to the -term by the address function on input bits. Gathen and Roche  show that for any non-trivial symmetric Boolean function, where is the largest prime below . (Notice that for symmetric functions, module degree can give a lower bound of degree.) Using currently best result on prime gaps , this gives an lower bound. On the other side, Gathen and Roche give a polynomial family with , and they propose Conjecture 2

below with a probabilistic heuristic argument:

###### Conjecture 2.

For any non-trivial symmetric Boolean function , .

#### Our Results.

In this paper, we extend many previous works by giving better lower bounds for . To be concrete, we focus on general Boolean functions over , and symmetric Boolean functions over where has at least two distinct prime factors. As we have already mentioned it, the gap between and can be arbitrarily large. Nevertheless, we claim that cannot be too small either. This begins with symmetric functions:

###### Theorem 1.

For any prime , positive integer , and non-trivial symmetric function with where , . The lower bound is tight.

In addition, Theorem 1 can be extended to general non-degenerated Boolean functions. We achieve this using an embedding technique from hypergraph Ramsey theory.

###### Theorem 2.

For any prime , positive integer , and non-degenerated function with sufficient large , . The lower bound is tight.

For any non-prime-power composite , the following lower bound on modulo- degree of symmetric functions can be obtained:

###### Theorem 3.

For any composite number with at least two different prime factors and any non-trivial symmetric Boolean function , .

Note that this bound approaches when and are also growing with . It improves the bound in . On the other hand, the next theorem shows that the lower bound in Theorem 3 cannot be larger than :

###### Theorem 4.

For any two different prime numbers and , there exists symmetric with arbitrarily large , such that .

## 2 Preliminaries

### 2.1 Boolean Functions and Polynomial Representation

An -bit Boolean function is a mapping from to . We say a Boolean function is non-trivial if it is not a constant function. A Boolean function is non-degenerated, if there does not exist any that satisfies, for all , Here means we fix the th bit of to be .

We say a polynomial represents a Boolean function over ring if for all . Actually, the representation over is unique, owing to the following fact.

###### Fact 1.

For any Boolean function , the unique polynomial

 ∑a∈{0,1}nf(a)n∏i=1((2ai−1)xi+1−ai)=:∑S⊆[n]αS∏i∈Sxi

represents over .

Analogously, the representation of over is unique too: is its representation. These facts allow the degree, as well as the modulo- degree, to be well-defined.

###### Definition 1.

The degree (resp., modolo- degree) of a Boolean function , denoted by (resp., ), is the degree of the polynomial representing over (resp., ).

The following facts are useful for analysis.

###### Fact 2.

For any Boolean function and integer ,

• if , then ;333It follows from Chinese Remainder Theorem.

• if , then ;

• furthermore, we have .

A Boolean function is symmetric if for any satisfying . Here is the number of s in . A symmetric can be written as a Mahler expansion of , i.e.,

Note that on any commutative ring , if the domain of is , then is uniquely determined. This can be shown by observing that (i) the multilinear representation of is unique due to the Möbius inversion presented above and (ii) there is a one-to-one correspondence between the multilinear representation and the univariate representation of a symmetric function.

We call the univariate representation of . To ease the notation, we sometimes still write when we refer to in the rest of the paper. In such case the ambiguity can be eliminated by the range of (a binary string or an integer).

###### Definition 2 (Based period of symmetric Boolean functions).

For a symmetric with the univariate representation , we say is -periodic if for all , implies . Define , the periodic of on base , as the minimal integer such that is a power of and is -periodic. (Note that it may be larger than .)

We give several examples to help reader understand this definition.

• and .

• where is any constant function.

• where if and only if .

### 2.2 MOD and Its Mahler Expansion Representation over Zpk

We look into a class of extended parity functions, weight modular functions, which is an indicator of whether the weight of the input is congruent to modulo .

###### Definition 3 (Weight modular functions).

For positive integer and , define a weight modular function as

 MODc,mn(x)={1if |x|≡c(modm),0otherwise.

As a prior work, Wilson researched the univariate representation of and showed the following result.

###### Theorem 5 (Wilson, ).

Given prime , positive integer , let .

• For -periodic symmetric , .

• For -periodic symmetric with Mahler expansion , holds for all .

• For all , .

For all , let be the Mahler expansion of . Obviously, it can also be represented by “shifted” Mahler expansion with coefficients of as . Thus, we get by Vandermonde convolution, which implies for all and leads to the following corollary.

###### Corollary 1.

Given prime , positive integer , let . For all and , .

Consider a special case over . Assume without loss of generality. Let be the Mahler expansion coefficient matrix of s satisfying the condition that, for and ,

 MODi,ptn(x)=pt−1∑j=0(Apt)j,i(|x|)j.

Note that such always exists since holds for all . It is feasible to set for all since

 pt−1∑j=0(pt−1−ipt−1−j)(|x|j)=(pt−1−i+|x|pt−1)={1if |x|≡i(modpt),0otherwise.

The selection of is unique due to the uniqueness of Mahler expansion. According to Lucas’s Theorem, since

 (pt−1−ipt−1−j)≡(∑t−1ℓ=0(p−1−jℓ)⋅pℓ∑t−1ℓ=0(p−1−iℓ)⋅pℓ)≡t−1∏ℓ=0(p−1−iℓp−1−jℓ)≡t−1∏ℓ=0(Ap)iℓ,jℓ(modp),

where and . As we have already mentioned, for any -periodic Boolean function , is a conversion matrix between and , i.e., where

is the Mahler expansion coefficient vector of

and is the univariate representation such that for all .

## 3 Lower Bound of degpk(f)

By identifying the degree of over , we show that the degree of all -periodic functions is constantly small since they can be spanned by . In Section 3.1, we prove that the degree of any -periodic (but not -periodic) function will not decrease too much from during the spanning, despite the cancellation of the high-order coefficients. By a Ramsey-type argument in Section 3.2, we further extend our lower bound to all non-degenerated Boolean function with sufficiently many input bits.

### 3.1 Symmetric Functions

First, we give a lower bound for -periodic functions. Note that all -periodic functions also have a upper bound of , and therefore our lower bound is tight.

###### Lemma 1.

For all non-trivial -periodic symmetric Boolean with , .

###### Proof.

Span by with , i.e., . Since is non-trivial, . The highest-order coefficients for all are the same, and moreover, the degree of the sum will not decrease because according to Theorem 5. ∎

Second, for functions with large -period, i.e, , also holds.

###### Lemma 2.

For all non-trivial symmetric Boolean function with and , holds.

###### Proof.

Set and . Since is -periodic, is spanned by with coefficient vector , or spanned by with coefficient vector . Let be the th Mahler expansion coefficient of . According to Theorem 5, for all and . So, we define reduced Mahler expansion coefficient for all and . Let be the matrix gathering the highest reduced coefficients of each where for and . Note that

 ⎡⎢ ⎢ ⎢ ⎢⎣~α(m−1)d⋯~α(m−1)d−m+1⋮⋱⋮~α(0)d⋯~α(0)d−m+1⎤⎥ ⎥ ⎥ ⎥⎦=⎡⎢ ⎢ ⎢⎣(1−m0)⋯(1−mm−1)⋮⋱⋮(00)⋯(0m−1)⎤⎥ ⎥ ⎥⎦⋅⎡⎢ ⎢ ⎢ ⎢⎣~α(0)d⋯~α(0)d−m+1⋱⋮~α(0)d⎤⎥ ⎥ ⎥ ⎥⎦

is full-rank over . Consequently, is full-rank and the dimension of the null space is . Define such that . Since , over for all . So, is the null space of . Note that is not -periodic and cannot be spanned by , i.e., , which implies and . ∎

Finally, when the -period of is so large that is larger than , the fact that is not -periodic implies a lower bound due to the counter-proposition of the following lemma.

###### Lemma 3.

Boolean function is -periodic if .

###### Proof.

Let and be the Mahler expansion over . Note that the value is zero or one. So, for all . Meanwhile, for any where , for all due to Lucas’s Theorem, which means is -periodic. ∎

We are ready to prove Theorem 1.

###### Proof of Theorem 1.

Assume towards contradiction there exists such that . According to Lemma 3, is -periodic where . Being non-trivial, is not -periodic. Thus, there exists such that is -periodic but not -periodic. If , due to Lemma 2. Otherwise, and due to Lemma 1. ∎

### 3.2 Non-Degenerated Functions

With the help of Ramsey theory on hypergraphs, the bound for symmetric functions can be applied to a wider class of Boolean functions — the non-degenerated functions.

###### Theorem 6 (Erdős and Rado, ).
 rk(s,t)≤2(rk−1(s−1,t−1)k−1),

where is Ramsey number on -uniform hypergraphs.

The theorem indicates the following property: any -edge-colored -uniform hypergraph on vertices has a monochromatic clique of size ,444. . where all hyperedges within have the same color. This property allows us to embed an -size symmetric function into any non-degenerated function , provided sufficiently large. Before continuing, we need to introduce the sensitivity of a Boolean function.

###### Definition 4.

Given a Boolean function and an input , we say a bit is sensitive if . Here is the string which differs from on the th bit. The sensitivity of on input is . The sensitivity of is then defined as .

Simon  proved that for any non-degenerated function , . In addition, we also find the following notation useful.

###### Definition 5.

Given positive integer , and where for all , define

 DOWN(S,x,k):={x⊕T∣T⊆S,|T|=k}.

Here, is the string which only differs from on all the th () bits.

###### Lemma 4.

For any non-degenerated function with sufficient large , there exists and such that is a non-trivial symmetric function.

###### Proof.

Pick such that . Define . W.l.o.g, assume . Define recursively as the maximum set such that and satisfies all has the same value. Next, we show the size of cannot be too small.

.

###### Proof.

Construct an edge-coloring for complete -hypergraph with the vertex set . For all input , color in black if . The other edges are colored in white. Let be the maximum monochromatic clique. According to Theorem 6,

 |St|=Ω(log∘(t−1)(|St−1|)).

Note that the inputs in have the same value, since it corresponds to a monochromatic clique. ∎

By applying Claim 1 inductively, . For any , let . Therefore, satisfies , and thus . Furthermore,

 r(n):=min{log∘t′(t′−1)/2+1(n),t′(n)}=ω(1).

Pick arbitrary -size subset of . We fix the function on the input for all bits in , i.e., let as an -variable Boolean function. Recall the definition of . For any where , let

 x′=x|[n]∖T←~x and y′=y|[n]∖T←~x.

Note that

 x′,y′∈DOWN(Sr(n),~x,|x|)⊆DOWN(S|x|,~x,|x|).

Thus, , namely, is symmetric. Let and be any input with weight . Then denote

 z′=z|[n]∖T←~x and w′=w|[n]∖T←~x.

If is a sensitive bit then and . Hence, is non-trivial because . ∎

Consequently, an -size symmetric function can be embedded into any non-degenerated function. It immediately leads to that, for any non-degenerated function with where is defined in Theorem 1, .

## 4 Lower Bounds of degpq(f) for Symmetric Functions

### 4.1 Proof of Theorem 3

Suppose contains at least two different prime factors and . To prove Theorem 3, the simple fact that will become crucial. To be concrete, and cannot be small simultaneously, due to the following two lemmas.

###### Lemma 5 (Periodicity Lemma, ).

Let be an -periodic and -periodic function on domain with and . Then is a constant function.

###### Lemma 6.

Let be a prime. For any non-trivial symmetric , .

For any non-trivial symmetric Boolean function and two different primes and , if , then the theorem is self-evident; otherwise, by Lemma 6, we have

 degpq(f)=max{degq(f),degq(f)}≥max{(1−1p)πp(f),(1−1q)πq(f)}.

On the other hand, since is not a constant function, by Lemma 5, we have . Combining the results above we get .

The only thing left here is why Lemma 6 holds. Before continuing, we need the following two lemmas.

###### Lemma 7.

For any prime , integers with and distinct satisfying , the following matrix is non-singular over ,

 ⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣(a0j)(a1j)⋯(akj)(a0j+1)(a1j+1)⋯(akj+1)⋮⋮⋱⋮(a0j+k)(a1j+k)⋯(akj+k)⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦.
###### Proof.

It is easy to see that

 diag⎛⎝(j+0)0–(a0j),…,(j+k)k––(akj)⎞⎠⋅S⋅⎡⎢ ⎢ ⎢ ⎢⎣(a0j)⋯(akj)⋮⋱⋮(a0j+k)⋯(akj+k)⎤⎥ ⎥ ⎥ ⎥⎦=⎡⎢ ⎢ ⎢⎣(a0−j)0⋯(ak−j)0⋮⋱⋮(a0−j)k⋯(ak−j)k⎤⎥ ⎥ ⎥⎦,

where is the second Stirling number matrix, i.e.,