Waring decompositions

SJÄLVSTÄNDIGA ARBETEN I MATEMATIK

MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET

Waring decompositions

av

Erik Jernqvist

2019 - No K3

MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET, 106 91 STOCKHOLM

Waring decompositions

Erik Jernqvist

Självständigt arbete i matematik 15 högskolepoäng, grundnivå Handledare: Samuel Lundqvist

2019

ERIK JERNQVIST

Abstract. A Waring decomposition of a homogeneous polynomial f is a sum of powers of linear forms expressing f The Waring rank is the smallest possible number of summands in such a decomposition. In this thesis we focus on decompositions of monomials. For decomposistions of monomials over the complex numbers the Waring rank is known. For decompositions over the reals or the rationals the Waring rank is not known, apart from two classes of monomials. For monomials with the smallest exponent equal to one and two-variable monomials, the Waring rank over the rationals and the reals are known and coincides. For these cases we give explicit constructions that yield minimal decompositions. For other monomials we examine the upper bounds for the ranks over the rationals and the reals. We also present some results for polynomials.

1. Introduction 1

2. Preliminaries 3

2.1. Explicit Waring decomposition overC 3

2.2. Waring decompositions overR 4

2.3. Upper bounds overQ 5

3. Comparison of e(A0× A¹× · · · × Aⁿ) and e(B0× B¹× · · · × Bⁿ). 9

4. Explicit formula overQ when d⁰= 1 10

5. The binary case overQ and R 12

5.1. A naive construction overQ 12

5.2. Other constructions 19

6. Asymptotic behavior of the upper bounds overR 23

7. Computations 23

8. Open problems 25

9. Appendix: Mathematica code 25

9.1. Equivalence classes 25

9.2. The naive construction for the binary case 26

9.3. Explicit construction overR 26

References 27

1. Introduction

This thesis deals with Waring decompositions of polynomials, and specifically mainly monomials. The decompositions are named after the 18th century British math- ematician Waring who proposed that for any k ∈ N there exists an s ∈ N such that any natural number could be written as a sum of at most s kth powers of natural numbers. This claim is known as Waring’s problem and was proven in 1909 by Hilbert. Lagrange had already proved 1707 that four squares suffices for any natural number, with 7 = 1²+ 1²+ 1²+ 2² being the smallest number that can not be written as the sum of three or less squares. Waring’s problem is related to two theorems connected to Fermat; his polygonal number theorem and of course his last theorem.

1

Instead of natural numbers we can consider polynomials and in the case of this thesis monomials especially. Though the references in this thesis are mainly from the last decade, already in the 19th century Sylvester did work on decompositions taken polynomials over C with a result proven in 1851 known as one of his eponymous theorem.

Given a fieldK and homogeneous polynomial f ∈ K[x⁰, x1, . . . , xn] of degree d, we define the Waring rank of f overK as the smallest natural number r such that

(1) f = X

1≤i≤r

bi(ai,0x0+ ai,1x1+· · · + a^i,nxn)^d,

We denotate this number as WK(f ) and call (1) a Waring decomposition of f over K. Note that there are fields where such a smallest r does not exist. For example overR we have the decomposition

xy = 1

4 (x + y)²− (x − y)²
,

but in Z/2Z we can not find any such decomposition of xy, since (ax + by)² = a²x²+ b²y²= ax²+ by² inZ/2Z. In general, in Z/pZ we have



 X

0≤i≤n

aixi





p

= X

0≤i≤n

a^p_ix^p_i = X

0≤i≤n

aix^p_i,

since p divides the multinomial coefficient

 p

q0, q1, . . . , qn



= _q0!q1^p!_!···qn!if q0, q1, . . . , qn6=

p. Because of this there are no decompositions of monomials of degree p overZ/pZ apart from pure powers.

Let us now consider decompositions of monomials in particular. If we let m = x^d₀⁰x^d₁¹· · · x^dnⁿ, then for the case when K = C it is known from [3] that

WC(m) = Y

1≤i≤n

(di+ 1).

ForK = R less is known. It was proven in [4] that if the smallest exponent is equal to one, then WR = WC. For the binary case, i.e. in the case of a two-variable monomial x^d₀⁰x^d₁¹, in [1] it was shown that WR(m) = d = d0+ d1 by using the Apolarity Lemma.

The Waring rank is not known if d0> 1 or we have more than two variables. Indeed, for the simplest possible example x²₀x²₁x²₁ it is not known if the decomposition

360x²₀x²₁x²₂= (x0+ x1+ x2)⁶+ (x0+ x1− x²)⁶+ (x0− x¹+ x2)⁶+ (x0− x¹− x²)⁶

−2[(x⁰+ x1)⁶(x0− x¹)⁶+ (x0+ x2)⁶(x0− x²)⁶(x1+ x2)⁶(x1− x²)⁶] +4(x⁶₀+ x⁶₁+ x⁶₂)

with its 13 terms is a minimal decomposition over the reals.

In this thesis the main focus will be put on examining the upper bounds for the Waring rank over monomials overQ (which also serve as upper bounds over R) and

explicit constructions over Q and R when the rank is known, i.e. d⁰ = 1 and the binary case.

2. Preliminaries

In this section we will present some previously established results.

2.1. Explicit Waring decomposition over C. It was first proven in [3] that

WC(m) = Y

1≤i≤n

(di+ 1),

but no explicit formula for such a decomposition was given there.

Theorem 1. [2, 2. Explicit expression for monomials as sums of powers] Consider a monomial x^d:= x^d₀⁰x^d₁¹· · · xn^dn and set d = d0+ d1+· · · + dⁿ and ζi be primitive (di+ 1)st root of unity. We have the following minimal decomposition

(2) x^d₀⁰x^d₁¹· · · x^dnⁿ= 1 C

X

0≤ai≤di

i=1,2,...,n

(ζ₁^a1· · · ζn^an)(x0+ ζ₁^a1x1+· · · + ζn^anxn)^d,

where

C =

 d

d0, . . . , dn



(d1+ 1)· · · (dⁿ+ 1) = d!

d0!· · · dⁿ!(d1+ 1)· · · (dⁿ+ 1).

Proof. Our argument follows the one given in [2]. We can start by noting that if the equality in (2) holds, then we have exactly Q

1≤i≤n(di+ 1) terms. Next, consider some monomial x^m = x^m0· · · x^mn, where m0+· · · + mⁿ = d. The coefficient of this monomial in the right-hand side of (2) is

Cm =

 d

m0, . . . , mn

 X

0≤ai≤di

i=1,...,n

ζ₁^(m1+1)· · · ζn^(mn+1),

which factors as

 d

m0, . . . mn

 X^d1

a1=0

(ζ₁^(m1+1))^a1

!

· · ·

dn

X

an=0

(ζ_n^(mn+1))^an

! .

Though it might not be primitive, ζ_i^mi+1 is still a (di+ 1)st root of unity, so we have

d1

X

ai=0

(ζ_i^mi+1)^ai=

( di+ 1, ζ_i^mi+1= 1, 0 otherwise.

Hence

(3) Cm=







1 C

d m0, . . . , mn

!

(d1+ 1)· · · (dⁿ+ 1), ζ₁^m1+1=· · · = ζn^mn+1= 1,

0 otherwise.

From (3) we see that in order for (2) to hold m = d ⇒ C^m = 1, as desired. All that remains is to show that m6= d ⇒ C^m= 0.

In order for Cm = 1 to hold, by (3) ζ_i^mi+1= 1 for 1≤ i ≤ n. Since ζⁱ is a primitive (di+ 1)st root, this means that mi+ 1 is a multiple of di+ 1 for 1≤ i ≤ n. Recall that m0+· · · + mⁿ = d0+· · · + dⁿ, so m0 ≤ d⁰. If for some i, mi 6= dⁱ, then mi+ 1≥ 2(dⁱ+ 1) and so m0≤ d⁰− (dⁱ+ 1) < 0, since d0≤ · · · ≤ dⁿ. Thus we have contradiction, since m0> 0. We can then conclude that Cm = 1 if and only

if m = d and else Cm= 0, so (2) holds. 

Example 1. Let us consider the monomial x²₀x³₁. Using Theorem 1 we can find the minimal decomposition

x²₀x³₁= 4

5! (x0+ x1)⁵+ i(x0+ ix1)⁵− (x⁰− x¹)⁵− i(x⁰− ix¹)⁵
.

2.2. Waring decompositions over R. In this case, there is no known explicit formula like the one for C. The following results can be found in [4].

Theorem 2. [4, Theorem 3.1] If m = x^d₀⁰· · · x^dnⁿ with 0 < d0≤ · · · ≤ dⁿ then WR(m)≤ 1

2d0

Y

0≤i≤n

(di+ d0).

Corollary 3. [4, Corrolary 3.2] If m = x0x^d₁¹· · · xn^dn thenWC(m) =WR(m).

Proof. SinceR ⊂ C we have WR(m)≥ WC(m). By Theorem 1 we have

WC(m) = Y

1≤i≤n

(di+ 1)

and if d0= 1 Theorem 2 implies that

WR(m)≤1 2

Y

0≤i≤n

(di+ 1) =WC(m).

Thus if d0= 1 thenWC(m) =WR(m). 

Proposition 4. [1, Proposition 4.4] If m = x^d₀⁰x^d₁¹ and 0 < d0≤ d¹thenWR(m) = d0+ d1.

2.3. Upper bounds overQ. Given in [5] we have a way to use the explicit formula overC given in Theorem 1 in order to find upper bounds over Q.

Consider the monomial y0· · · y^d−1. We can directly use (2) to find a decomposition (that may or may not be minimal) over Q, since for all i we have dⁱ = 1, so the (di+ 1)th roots of unity will be the second root of unity−1. Hence ζⁱ =−1 ∈ Q for all i. We then get

(4) y0· · · y^d−1 = 1 C

X

0≤bi≤1 i=1,...,d−1

(−1)^b1· · · (−1)^bd−1(y0+ (−1)^b1y1· · · (−1)^bd−1yd−1)^d,

where

C =

 d

1, . . . , 1



(1 + 1)· · · (1 + 1) = d! · 2^d−1.

Now consider a monomial x₀^d0x^d₁¹· · · xn^dn such that d0+· · · + dⁿ = d and let

Sj :=

( −1 j =−1,

P

0≤i≤jdi j≥ 0.

Now we can do the simple variable substitution

(5) yi= xj, Sj−1< i≤ S^j. Define Ai as

Ai=

( {−dⁱ+ 2,−dⁱ+ 4 . . . , di− 2, dⁱ} i = 0, {−dⁱ,−dⁱ+ 2 . . . , di− 2, dⁱ} i > 0.

Furthermore, let Ω = A0× A¹× · · · × Aⁿ and let x∼ y be the equivalence relation on Ω such that x ∼ y if there exists λ 6= 0 such that λx = y. Denote e(Ω) the number of non-zero equivalence classes on Ω with respect to∼.

Lemma 1. [5, Lemma 1] Let x^d₀⁰x^d₁¹· · · x^dnⁿ be a monomial of degree d > 1 with 1≤ d⁰≤ · · · ≤ dⁿ then

WQ(x^d₀⁰x^d₁¹· · · xn^dn)≤ e(A⁰× A¹× · · · × Aⁿ).

Proof. From the variable substitution in (5) it follows that x^d₀⁰x^d₁¹· · · x^dnⁿ can be written as a linear combination of the 2^d−1 terms found in (4). After substitution these terms are of the form

(x|0± x{z⁰· · · x⁰}

d0times

±x¹± x¹· · · x¹

| {z }

d1times

· · · ±x| ⁿ± x{zⁿ· · · xⁿ}

dntimes

)^d.

We can now notice that the set

{x|⁰± x{z⁰· · · x⁰}

d0times

±x¹± x¹· · · x¹

| {z }

d1times

· · · ±x| ⁿ± x{zⁿ· · · xⁿ}

dntimes

}

is equal to set

{a⁰x0+· · · + aⁿxn: (a0, . . . , an)∈ Ω}.

Moreover, it is clear that if (a0x0+· · · + aⁿxn) and (b0x0+· · · + bⁿxn) are linearly dependent if (a0, . . . , an)∼ (b⁰, . . . , bn). Hence, the Waring rank of our monomial

is at most e(A0× A¹× · · · × Aⁿ). 

We will use this result to find a upper bound for Waring rank overQ.

Lemma 2. If A⁰₀ is comprised of all the non-negative elements of A0, then e(A0× A¹× · · · × Aⁿ) = e(A⁰₀× A¹× · · · × Aⁿ).

Proof. We note that if (a0, . . . , an)∈ Ω and (a⁰, . . . , an)6∈ A⁰0× · · · × Aⁿ then this would be because a0 6∈ A⁰0. However, since a0 ∈ A⁰ we have −a⁰ ∈ A⁰0, implying that (−a⁰, . . . ,−aⁿ) ∈ A⁰0× · · · × Aⁿ. Since (a0, . . . , an) ∼ (−a⁰, . . . ,−aⁿ) the

number of equivalence classes stays the same. 

Theorem 5. [5, Theorem 2] If m = x₀^d0x^d₁¹· · · xn^dn and 0 > d0≤ d¹≤ · · · dⁿ then WQ(m)≤ (d0+ 1)· · · (dⁿ+ 1)

2 .

Proof. The proof below was found by the author. By Lemma 1 it suffices to show that e(Ω)≤ ^{(d0+1)···(d}2 ⁿ⁺¹⁾.

Assume that d0 is odd. Then by Lemma 2 we have

e(Ω)≤ |A⁰0||A¹| · · · |Aⁿ| =(d0+ 1)· · · (dⁿ+ 1)

2 ,

since|A⁰0| = ^d02⁺¹ and|Aⁱ| = dⁱ+ 1 for i = 1, . . . , n.

Now assume that d0 is even. Then by the previous reasoning

(6) e(Ω)≤ |A⁰0||A¹| · · · |Aⁿ| = (d0+ 2)(d1+ 1)· · · (dⁿ+ 1)

2 ,

since|A⁰| = d⁰+ 2. In (6) we have counted ^d1(d2+1)₂^···(dn+1) cases where a0= 0 and a1< 0. By the same line of reasoning as in Lemma 2 we can safely exclude these.

Subtracting ^d1(d2+1)₂^···(dn+1) from (6) we get

e(Ω)≤ (d0+ 2)(d1+ 1)· · · (dⁿ+ 1)

2 −d1(d2+ 1)· · · (dⁿ+ 1)

2 = (d0+ 1)· · · (dⁿ+ 1)

2 .

 From Theorem 5 we can easily get that the rank overQ will coincide with the rank overC if d⁰= 1.

Corollary 6. Let m = x₀^d0x^d₁¹· · · xn^dn with d0 ≤ d¹ ≤ · · · ≤ dⁿ. If d0 = 1 then WC(m) =WR(m) =WQ(m).

Proof. Note that if d0 = 1, the upper bound from Theorem 5 and the one from 2 coincide. Since this upper bound is exactly WC(m) and Q ⊂ R ⊂ C we have

WC(m) =WR(m) =WQ(m) if d0= 1. 

An alternative way of proving that a monomial x^d₀⁰x^d₁¹ can be written as a linear combination of d0+ d1 powers of linear forms is by noticing that the construction overR given in [1] can be modified to hold over Q as well, see [5].

Example 2. Consider the monomial x³y⁴z⁵. Then we have

Ω ={−3, −1, 1, 3} × {−4, −2, 0, 2, 4} × {−5, −3, −1, 1, 3, 5}.

By Lemma 2 we only need to count the number of equivalence classes of

Ω⁰:={1, 3} × {−4, −2, 0, 2, 4} × {−5, −3, −1, 1, 3, 5}.

From this we get the upper bound

WQ(x³y⁴z⁵)≤ e(Ω) = e(Ω⁰)≤ |Ω⁰| =(3 + 1)(4 + 1)(5 + 1)

2 = 60.

We can improve this bound by noting that (1, 0, 1) ∼ (3, 0, 3) and (1, 0, −1) ∼ (3, 0,−3). Then

WQ(x³y⁴z⁵)≤ e(Ω⁰) = 58.

Theorem 7. [5, Theorem 4] If m = x^d₀⁰x^d₁¹· · · x^dnⁿ then WQ(m)≤ min

(d0+ 1)· · · (dⁿ+ 1)

2 , (d1+ d0)· · · (dⁿ+ d0)

 .

Proof. By Theorem 5 we have

WQ(m)≤ (d0+ 1)· · · (dⁿ+ 1)

2 ,

and by Theorem 2 we have

WQ(m)≤ (d¹+ d0)· · · (dⁿ+ d0).

 There is also another upper bound given by [5], which as we will prove later is an improvement of Theorem 5 and Theorem 7. First we will state a helpful lemma.

Lemma 3. [5, Lemma 5] Let m = x₀^2d0x^2d₁ ¹· · · xn^2dn with 1≤ d⁰≤ · · · dⁿ then WQ(m)≤ e(B⁰× B¹× · · · × Bⁿ),

where B0= [−d⁰+ 1, d0] and Bi= [−dⁱ, di].

Proof. If x ∼ y, then there is some non-zero λ such that λx = y. We can now conclude our proof by noting that we only need to multiply with a constant factor

1

2 to change from A0×···×Aⁿ to B0×···×Bⁿ and that it is clear that λ^x₂ = ^y₂.  Theorem 8. [5, Theorem 6] If m = x₀^d0x^d₁¹· · · xn^dn then WQ(m) ≤ e(B⁰× B¹×

· · · Bⁿ), where

B0=

( 1,d0

2

 d0 odd, [0,^d₂⁰] d0 even,

Bi =

( −_d0

2

,_d0

2

\{0} d0 odd,

[−^d2⁰,^d₂⁰] d0 even, i > 0.

Proof. Consider the monomial M := x^e₀⁰· · · x^enⁿ, where

ei =

( di di even, di+ 1 di odd,

so every ei is even. LetQ[y⁰, . . . , yn] be the ring of differential operators acting on Q[x⁰, . . . , xn]. Suppose that 2≤ d⁰≤ · · · ≤ dⁿ and set

fi=

( 0 di even, 1 di odd. . Then

y₀^f0· · · y^fnⁿM =



 Y

i∈{i:fi=1}

di



 m.

If M = c1l^d₁+· · · c^sl^d_s is a Waring decomposition, then

Q 1

i∈{i:fi=1}di

y₀^f0· · · yn^fnc1l₁^d+· · · c^sl^d_s

is a Waring decomposition of m. From Lemma 1, Lemma 2 and Lemma 3 and Theorem 5, we have that

WQ(M )≤ e(B^M0 × · · · × Bn^M),

where B₀^M = [0,^e₂⁰] and B_i^M = [−^e2ⁱ,^e₂ⁱ]. We note that if diis odd and the coefficient of xi in lj is zero, then y₀^f0· · · yn^fn· l^dj = 0. So for the Waring decomposition of m we can safely assume that bi6= 0 if dⁱ is odd. So we have

WQ(m)≤ e(B⁰× B¹× · · · × Bⁿ),

Bi=

( B_i^M di even, B_i^M\{0} dⁱ odd,

which is just another way of writing Bi, i = 0, 1, . . . , n in the statement of the

theorem. 

3. Comparison of e(A0× A¹× · · · × Aⁿ) and e(B0× B¹× · · · × Bⁿ).

In this section we will show that the upper bound of Theorem 8 is always at least as good as the one found in 5.

Theorem 9. If m = x^d₀⁰x^d₁¹· · · xn^dn then e(A0× A¹× · · · × Aⁿ) and e(B0× B¹×

· · · × Bⁿ) are both upper bounds forW(m)Q and

e(B0× B¹× · · · × Bⁿ)≤ e(A⁰× A¹× · · · × Aⁿ) for all monomials.

Proof. Set

Φ(x) =

( _x

2 x even,

x+sgn(x)

2 x odd, where sgn(x) :=







−1 x < 0, 0 x = 0, 1 x > 0 and

bi= Φ(ai), 0≤ i ≤ n.

By Theorems 5 and 8 we know thatW(m)Q≤ e(B⁰× B¹× · · · × Bⁿ) andW(m)Q≤ e(A0×A¹×· · ·×Aⁿ). What remains to show is that e(A0×···×Aⁿ)≥ e(B⁰×···×Bⁿ) for all A0× A¹× · · · × Aⁿ and B0× B¹× · · · × Bⁿ. We do this by showing that for every integer α such that α /∈ {0, 1} and

α(a0, . . . , an)∈ A⁰× A¹× · · · × Aⁿ, there is a injective function β(α) such that β(α) /∈ {0, 1} and

β(α)(b0, . . . , bn)∈ B⁰× B¹× · · · × Bⁿ.

Let us first assume that all diare even, then β(α) = α satisfies our conditions and by using Lemma 3 we get that e(A0× A¹× · · · × Aⁿ) = e(B0× B¹× · · · × Bⁿ).

Next let us assume that there is at least one odd exponent. We want show that

α(a0, . . . , an)∈ A⁰×A¹×···×Aⁿ =⇒ Φ(α)(Φ(a⁰), . . . , Φ(an))∈ B⁰×B¹×···×Bⁿ, i.e. we choose β(α) = Φ(α). First we need to prove that Φ(α) is injective. Let di

be the smallest odd exponent. By assumption, αai∈ Aⁱ, which implies that α≤ dⁱ and that α is odd. Thus α∈ Aⁱ. Hence, α will always be odd in this case, so Φ(α) will be injective, since Φ(x) = Φ(y)⇒ |x − y| = 1. We note that

α(a0, . . . , an)∈ A⁰× · · · × Aⁿ ⇐⇒ |α||a^j| ≤ d^j and

Φ(α)(Φ(a0), . . . , Φ(an))∈ B⁰× · · · × Bⁿ⇐⇒ |Φ(α)Φ(a^j)| ≤ Φ(d^j).

In the case when dj is even, the condition|Φ(α)Φ(a^j)| ≤ Φ(d^j) is equivalent to

|α| + 1 2

|a^j| 2 ≤ dj

2

⇐⇒ |α| + 1

2 |a^j| ≤ d^j. As ^|α|+1₂ ≤ |α| for every odd α, we have, for even d^j that

|α||a^j| ≤ d^j =⇒ |Φ(α)Φ(a^j)| ≤ Φ(d^j).

In the case when dj is odd,|Φ(α)Φ(a^j)| ≤ Φ(d^j) is equivalent to

|α| + 1 2

|a^j| + 1

2 ≤ dj+ 1 2

⇐⇒ |α| + 1

2 (|a^j| + 1) ≤ d^j+ 1.

We note that ^|α|+1₂ (|a^j| + 1) ≤ ^|α|+12 |a^j| ≤ |α||a^j| and d^j < dj+ 1. From this we get

|α||a^j| ≤ d^j =⇒ |Φ(α)Φ(a^j)| ≤ Φ(d^j) when dj is odd. This completes our proof.



4. Explicit formula over Q when d⁰= 1

We have seen that in general we can only obtain upper bounds for the Waring rank of monomials over the reals or the rationals. However from Corollary 6 we know that if d0= 1 then the ranks overC and R coincide and from Corollary 6 to that over Q. We will derive a explicit formula for this case, which will serve as a constructive proof of Corollary 6.

Proposition 10. If m = x0x^d₁¹· · · x^dnn then (7)

m = 1 C

X

0≤ai≤di

i=1,...,n

d1

a1



···

dn

an



(−1)^a1···(−1)^an(x0+(d1−2a¹)x1+···+(dⁿ−2aⁿ)xn)^d,

where C = d!· 2^d−1. Proof. By (4) and (5) we have

m = 1 C

X

0≤bi≤1 i=1,...,d−1

(−1)^b1· · · (−1)^bd−1(x0±x| ¹± · · · ± x{z ¹}

d1times

· · · ±x| ⁿ± · · · ± x{z ⁿ}

dntimes

)^d.

Now let

ai= X

S_i−1<j≤Si

bj.

Summing over aj= r for some j and r we get X

0≤bi≤1 aj=r i=1,...,d−1

(−1)^b1· · · (−1)^bd−1(x0± x¹± · · · ± xⁿ)^d

= X

0≤bi≤1 aj=r i=1,...,d−1

(−1)^b1· · · (−1)^r· · · (−1)^bd−1(x0± x¹± · · · + (d^j− 2r)x^j· · · ± xⁿ)^d.

We can now note that

dj

r



of the terms in (4) will satisfy aj = r. If we let Ij= [1, d− 1]\[S^j−1+ 1, Sj] we can rewrite the above sum as

X

0≤bi≤1 aj=r

i∈Ij

dj

r



(−1)^b1· · · (−1)^r· · · (−1)^bd−1(x0± x¹± · · · + (d^j− 2r)x^j· · · ± xⁿ)^d.

If we sum over 0≤ a^j ≤ d^j we obtain

m = X

0≤bi≤1 aj=0,...,dj

i∈Ij

dj

aj



(−1)^b1· · · (−1)^aj· · · (−1)^bd−1(x0±x¹±· · ·+(d^j−2a^j)xj· · ·±xⁿ)^d.

If we do this for all 1≤ j ≤ n, we obtain (7). 

Example 3. Let m = x0x²₁x³₂ and ω = e^2πi3 , i.e. a cubic root of unity. Using Theorem 1 we can find the minimal decomposition

m = 1 6!



− (x⁰+ x1− x²)⁶− ω² x0+ ω²x1− x^2
6

− ω²(x0+ ωx1− x²)⁶

−i(x⁰+x1−ix²)⁶−iω² x0+ ω²x1− ix²
6

−iω (x⁰+ ωx1− ix²)⁶+i(x0+x1+ix2)⁶ +iω² x0+ ω²x1+ ix2
6

+iω (x0+ ωx1+ ix2)⁶+(x0+x1+x2)⁶+ω² x0+ ω²x1+ x2
6

+ ω (x0+ ωx1+ x2)⁶

 .

Furthermore we can use Proposition 10 to find a minimal decomposition over Q that is also minimal overC. This decomposition is

m = 1

6!· 2⁵



−(x⁰−2x¹−3x²)⁶−(x⁰+2x1−3x²)⁶+(x0+2x1+3x2)⁶+(x0−2x¹+3x2)⁶ +3[(x0− 2x¹− x²)⁶+ (x0+ 2x1− x²)⁶− (x⁰− 2x¹+ x2)⁶− (x⁰+ 2x1+ x2)⁶]

+2[(x0− 3x²)⁶− (x⁰+ 3x2)⁶] + 6[(x0+ x2)⁶− (x⁰− x²)⁶]

 .

Clearly these two decompositions are distinct and are not trivial variations of each other.

5. The binary case overQ and R

In this section we will find explicit Waring decompositions for the binary case, i.e.

when we have a monomial with exactly two variables.

Consider some two-variable monomial x^d0y^d1, where d0, d1> 0. Then from Theo- rem 4 its rank overR is d⁰+ d1. From Corollary 6 we have that its rank overQ is d0+ d1 as well.

We can note that the vector space of homogeneous two-variable polynomials of degree d has dimension d + 1 with an obvious basis consisting {x^d, x^d−1y, . . . , y^d}.

So, any element in this vector space can be written as a linear combination of at most d + 1 terms from some set of linearly independent elements. Note that the monomial x^d0y^d1, where d0, d1 > 0 and d0 + d1 = d is a element of this vector space. From the results mentioned above we can see that x^d0y^d1 can be written as a linear combinations of d (rather than d + 1) powers of linear forms when working over the rationals. In the following sections we will present explicit constructions overQ and R.

5.1. A naive construction overQ. From Proposition 4 we know that W^R(x^d0y^d1) = d0+ d1. In this section we will present a construction overQ. For small d⁰, d1we can find a minimal decomposition over Q directly by using the construction from Section 2.3. By grouping together the 2^d−1 terms we can write

(8)

x^d0y^d1 = 1 2^d−1d!

X

0≤a1≤d1

0≤a0≤d0−1

d0− 1 a0

d1

a1



(−1)^a0(−1)^a1((d0−2a⁰)x+(d1−2a¹)y)^d.

In particular,we get

x²y³= 1

2⁴· 5! −3(2x + y)⁵+ 3(2x− y)⁵− 3(2x + 3y)⁵+ (2x− 3y)⁵− 480y⁵
. In order to find minimal decompositions over Q we will generalize the decomposi- tions we can find for small exponents. Let us first introduce our construction and prove its minimality, assuming it exists.

Lemma 4. If m = x^d0y^d1 and d0, d1 > 0. Assume there exists {γ^a}^a and {δ^b}^b such that

(9)

m = X

a∈U

γa((ax+f y)^d+(−1)^d0(−ax+fy)^d)+ X

b∈V \{f}

δb((ex+by)^d+(−1)^d1(ex−by)^d),

where

U =

( {0, 2, . . . , d⁰} d⁰ even,

{1, 3, . . . , d⁰} d0 odd, V =

( {0, 2, . . . , dⁱ} d¹ even, {1, 3, . . . , dⁱ} d1 odd,

e =

( 2 d0 even,

1 d0 odd, , f =

( 2 d1 even, 1 d1 odd.

Then (9) is a minimal decomposition of m over Q.

Proof. If d0 and d1 are both odd, then|U| = ^d02⁺¹ and|V | = ^d12⁺¹. Since we have two summands for each element in U and V\{f} we obtain

WQ(m)≤ 2(|U| + |V \{f}|) = 2

d0+ 1

2 +d1+ 1 2 − 1



= d0+ d1.

If d0 is even, then |U| = ^d2⁰ + 1 and if d1 is even then |V | = ^d2¹ + 1. Since 2 ^d₂ⁱ + 1

= 2 ^di₂⁺¹+

+ 1 it seems that we will have an extra summand in this case. However we can observe that if d0 is even then 0∈ U, so we will have a pair of summands of the form f^dx^d+ (−1)^d0f^dx^d, which simplifies to 2f^dx^d. Similarly if d1 is even then, 0∈ V , so we will have e^dy^d+ (−1)^d1e^dy^d= 2e^dy^d. 

Theorem 11. If m = x^d0y^d1 and d0, d1 > 0, then there exists {γ^a}^a and {δ^b}^b such that

(10)

m = X

a∈U

γa((ax+f y)^d+(−1)^d0(−ax+fy)^d)+ X

b∈V \{f}

δb((ex+by)^d+(−1)^d1(ex−by)^d)

is a minimal decomposition of m over Q.

Proof. By Lemma 4 we know that the decomposition will be minimal if it exists.

So we only need to prove that such a decomposition does exist. We can make the observation that

(ux + f y)^d+ (−1)^d0(−ux + fy)^d) = 2 X

0≤i≤d i≡d0 mod 2

d i



uⁱxⁱf^d−iy^d−i

and

(ex + vy)^d+ (−1)^d1(ex− vy)^d= 2 X

0≤i≤d i≡d0 mod 2

d i



eⁱxⁱv^d−iy^d−i.

Observe that if (10) is a valid Waring decompostion, then the coefficient of x^αy^β is non-zero if and only if α = d0 and β = d1. Using this fact we set up a system of equations and solve it for {γ^a}^a and{δ^b}^b.

Now let u and v be the smallest elements of U and V , respectively. Also let w be the smallest element of V\{f}. Furthermore, let

M=2

          

d u
uufd−ud u
(u+2)ufd−u···d u
d^{u 0}fd−ud u
euwd−u···d u
eu(d1−2)d−ud u
eudd−u 1 d u+2
uu+2 fd−u−2d u+2
(u+2)u+2 fd−u−2 ···d u+2
du+2 0fd1−u−2d u+2
eu+2 wd−u−2 ···d u+2
eu+2 (d1−2)d−u−2d u+2
eu+2 dd−u−2 1

. . . . . . . .. . . . . . . . .. . . . . . .

d d0
ud0fd1d d0
(u+2)d0fd1···d d
dd0 0fd1d d0
ed0dd1 1···d d0
ed0(d1−2)d1d d0
ed0dd1 1

. . . . . . . .. . . . . . . . .. . . . . . .

d d−v
ud−vfvd d−v
(u+2)d−vfv···d d−v
dd−v 0fvd d−v
ed−vwv···d d−v
ed−v(d1−2)d1d d−v
ed−vdd1 1

          

,

where we take elements of U0in ascending order from u to d0and then the elements of V\{f} in ascending order from v to d¹. Now set

C =















 γu

γu+2

... γd0

δw

... δd1−2

δd1

















and R =

















 0 0 ... 0 1 0 0 ... 0



















such that C· R = γ^d0.

Note that M C represents the coefficients on the left-hand side of (10) and R rep- resents the coefficients of m. Hence there is a decomposition of the form in (10) if and only if there is a C such M C = R. Since we want to solve the system for C, we want M to be invertible. We will show that M is indeed invertible by showing that its determinant is non-zero.

If d0is even, then the leftmost column of M will be of the form







 f^d−u

0 0 ... 0









and if d1

is even the |U| + 1-th column will be of the form







 0 0 ... 0 e^d−v









, since u = 0 or v = 0,

respectively. Let M⁰ be the minor of M when expanded along a column of this type, or in the case when both exponents are even; when expanded along the two columns of this type. For the case when both exponents are odd just use M⁰= M . For all these cases we will have that |M| 6= 0 if and only if |M⁰| 6= 0.

Now let M⁰⁰ be the matrix where we have taken ¹₂M⁰ and divided each row with the binomial coefficient that appears in it and then divided each column with the element in the first row of the same column. We then get

M”=

       11···111···1 e2 f−2 (e+2)2 f−2 ···d^{2 0}f−2 e2 w0−2 e2 (w0 +2)−2 ···e2 d−2 1 e4f−4(e+2)4f−4···d^{4 0}f−4e4w0−4e4(w0+2)−4···e4d−4 1

. . . . . . . .. . . . . . . . . . . .. . . .

ed−u−vfu+v−d(e+2)d−u−vfu+v−d···dd−u−v 0fu+v−ded−u−vw0u+v−ded−u−v(w0+2)u+v−d···ed−u−vdu+v−d 1

      

, wherew’isthesmallestelementofV\{0,f}.Wecanseethat M”=

           11···111···1 e2 f2(e+2)2 f2···d2 0 f2e2 w02e2 (w0+1)2···e2 d^{2 1} e2 f22 (e+2)2 f22 ··· d2 f22 e2 w022 e2 (w0+2)22 ··· e2 d^{2 1}2

. . . . . . . .. . . . . . . . . . . .. . . .

 e2 f2d−u−v 2 (e+2)2 f2d−u−v 2 ··· d2 f2d−u−v 2 e2 w02d−u−v 2 e2 (w0+2)2d−u−v 2 ··· e2 d^{2 1}d−u−v 2

          .

From the equality above we see that M⁰⁰is a Vandermonde matrix. Then we know that

(11) |M⁰⁰| =Y

i<j

(M_2,j⁰⁰ − M2,i⁰⁰ ).

By construction, we also know that|M| = 0 ⇐⇒ |M⁰⁰| = 0. From (11) we can see that

|M| 6= 0 ⇐⇒ (i6= j ⇒ M2,i⁰⁰ 6= M2,j⁰⁰ ).

Since all elements in U, V are non-negative we can write

u²₀ v²₀ = u²₁

v²₁ ⇐⇒ u0

v0

=u1

v1

, u0, u1∈ U⁰, v0, v1∈ U¹.

Then if we use the equivalence relation∼, we have u0

v0 = u1

v1 ⇐⇒ (u⁰, v0)∼ (u¹, v1).

We can note that in the second row of M⁰⁰we do not have elements corresponding to all elements in U× V , but only corresponding to all elements of

Ψ := ((U\{0}) × {f})a

({e} × (V \{0, f})).

From this we can see that elements of Ψ either has the form (a, f ) or (e, b), where a∈ (U\{0}) and b ∈ (V \{0, f}). It is easy to show that elements of the form (a, f) are pairwise linear independent, since

(u0, f ) = λ(u1, f )⇒ λ = 1 ⇒ u⁰= u1.

Because the same holds true for elements of the form (e, b) all that remains is to show that

(12) (u, f )∼ (e, v) ⇒ (u, f) = (e, v).

From the definition of ∼ we have

(a, f )∼ (e, b) ⇒ ∃λ 6= 0 :

( λa = e λf = b .

Since a∈ (U\{0}), λu = e implies λ ≤ 1. Similarly λf = b implies λ ≥ 1 so λ = 1, which proves (12).

Thus |M⁰⁰| 6= 0 which implies |M| 6= 0. Therefore we can find C, using C =

M⁻¹R.