uniqueness theorems for holomorphic curves with hypersurfaces of fermat–waring type

Complex Anal. Oper. TheoryDOI 10.1007/s11785-014-0367-1

Complex Analysisand Operator Theory

Uniqueness Theorems for Holomorphic Curveswith Hypersurfaces of Fermat–Waring Type

Ha Huy Khoai · Vu Hoai An · Le Quang Ninh

Received: 10 September 2013 / Accepted: 20 February 2014© Springer Basel 2014

Abstract In this paper, we establish uniqueness theorems for holomorphic mappingsfrom C to P N (C) for the case where the targets are not hyperplanes, but hypersurfacesof Fermat–Waring type.

Keywords Holomorphic curves · Uniqueness theorems

Mathematics Subject Classification (1991) Primary 32H02; Secondary 32H30

1 Introduction

In 1926, Nevanlinna proved the following result.

Communicated by Lawrence Zalcman.

The work was supported by a NAFOSTED grant.

H. H. Khoai (B)Institute of Mathematics, VAST, 18 Hoang Quoc Viet, Hanoi 10307, Vietname-mail: [email protected]

H. H. KhoaiThang Long University, Hanoi, Vietnam

V. H. AnHai Duong College, Hai Duong, Vietname-mail: [email protected]

L. Q. NinhThai Nguyen University of Education, Thai Nguyen, Vietname-mail: [email protected]

H. H. Khoai et al.

Theorem Let f, g be two non-constant meromorphic functions such that for fivedistinct values a1, a2, a3, a4, a5 in C ∪ {∞} we have f (x) = ai ⇔ g(x) = ai , i =1, 2, 3, 4, 5. Then f ≡ g.

Since that time the problem has been studied intensively and generalized in severaldirections.

In 1975 Fujimoto showed that for two linearly nondegenerate meromorphic func-tions f and g of P

N (C), if they have the same images counted with multiplicitiesfor 3N + 2 hyperplanes in general position in P

N (C), then f ≡ g. Many authorsgeneralized to the cases where either the require numbers of hyperplanes are smaller,or the hypothesis of nondegeneracy is avoided, or the multiplicities are truncated bysome k, or even ingnored (see [1–7]).

Some interesting results have been received also for the case where the hyperplanesare non constant, but moving ([8–10]). Recently in [3] the authors extended the resultsfor the case where the family of hyperplanes depends on the meromorphic mappings.

Another direction of generalizations is replacing the target hyperplanes by hyper-surfaces (see [1,9,11–13]).

In this paper, we establish uniqueness theorems for holomorphic mappings fromC to P

N (C) to the case where the targets are not hyperplanes, but hypersurfaces ofFermat–Waring type.

The hypersurfaces of Fermat–Waring type considered in this paper are of the fol-lowing class.

Let q, d, m, n be positive intergers, m < n, and let given q linear forms:

Li = αi,1z1 + αi,2z2 + · · · + αi,N+1zN+1, i = 1, 2, . . . , q.

Consider the following homogeneous polynomials:

Ti = Lni+1 − ai Ln−m

i+1 Lm1 + bi Ln

1; ai , bi �= 0; i = 1, 2, . . . , q − 1,

T = T d1 + T d

2 + · · · + T dq−1. (1.1)

Denote by X the hypersurface of Femat–Waring type in PN (C), which is defined by

the equation

T (z1, z1, . . . , zN+1) = 0.

For a holomorphic map f from C to PN (C) we denote by ν f (X) the pull-back of

the divisor X in PN (C) by f .

We shall prove the following uniqueness theorem.

Theorem 1 Let f and g be two non-degenerate holomorphic mappings from C toP

N (C). Let X be the Fermat–Waring hypersurface defined as above. Assume that:1/ ν f (X) = νg(X),

2/ n ≥ 2m + 9, (n, m) = 1, m ≥ 2, q > n; d ≥ (2q − 3)2,

Uniqueness Theorems for Holomorphic Curves

3/ b2di �= bd

j bdl with i �= j, i �= l.

Then f ≡ g.

2 Preliminaries

Let f be a non-constant holomorphic function on C. For every z0 ∈ C, expanding faround z0 as f = ∑

ai (z − z0)i , we define

ν f (z0) = min{i : ai �= 0}, n f (r) =∑

|z|≤r

ν f (z),

and for each a ∈ C, set n f (a, r) = n f −a(r).

Let r > 1, and for each a ∈ C, we define

N f (a, r) =r∫

1

n f (a, x)

xdx .

For l a positive integer, set

Nl, f (a, r) =r∫

1

nl, f (a, x)dx

x,

where

nl, f (a, r) =∑

|z|≤r

min{ν f −a(z), l}.

Now let f = f1

f2be a non-constant meromorphic function on C, where f1, f2 be

holomorphic functions on C having no common zeros. For a point a ∈ C and everyz ∈ C and for l a positive integer, we define

ν f −a(z) = ν f1−a f2(z), N f (a, r) = N f1−a f2(0, r), N f (∞, r) = N f2(0, r),

Nl, f (a, r) = Nl, f1−a f2(0, r), Nl, f (∞, r) = Nl, f2(0, r).

As usually, set

m f (∞, r) = 1

2π

2π∫

0

log+ | f (reiθ )|dθ,

T f (r) = N f (∞, r) + m f (∞, r).

H. H. Khoai et al.

Now let f be a holomorphic curve from C to PN (C). For an arbitrary fixed homoge-

neous coordinate system (z1 : · · · : zN+1) in PN (C) we take a reduced representation

of f : f̃ = ( f1 : · · · : fN+1). Set

T f (r) = 1

2π

2π∫

0

log || f̃ (reiθ )||dθ − log || f̃ (0)||.

Let H be a hypersurface of PN (C) such that the image of f is not contained in H ,

and H is defined by the equation F = 0. Set

N (H, r) = NF◦ f̃ (0, r), Nk, f (H, r) = Nk,F◦ f̃ (0, r).

For every z ∈ C set

ν f (H, z) = νF◦ f̃ (z),

and

ν f (H) = νF◦ f̃ .

Similarly, for the hypersurface X defined in the hypothesis of Theorem 1 we set:

ν f (X, z) = νT ◦ f̃ (z).

Therefore, the assumption 1/ in Theorem 1 means that

νT ◦ f̃ = νT ◦g̃.

In other words, if we set

E f (X) = {z ∈ C : T ◦ f̃ (z) = 0 counting multiplicities},Eg(X) = {z ∈ C : T ◦ g̃(z) = 0 counting multiplicities},

we have E f (X) = Eg(X).

Note that if f is a holomorphic curve from C to PN (C) and if f̃ = ( f1 : · · · : fN+1)

and h̃ = (h1 : · · · : hN+1) are two reduced representations of f , then there existsan entire funtion without zeros c such that fi = chi for all i . Therefore, the abovedefinitions are well-defined.

The following lemmas were proved in [14] (see also [15]).

Lemma 2.1 Let f be a non-constant meromorphic function on C and let a1, a2, . . . , aq

be distinct points of C ∪ {∞}. Then

(q − 2)T f (r) ≤q∑

i=1

N1, f (ai , r) + S f (r),


where S f (r) = 0(T f (r)) for all r, except for a set of finite Lebesgue measure.

Lemma 2.2 Let f be a non-constant meromorphic function on C and let a1, a2, . . . , aq

be distinct points of C ∪ {∞}. Suppose either f − ai has no zeros, or f − ai haszeros, in which case all the zeros of the functions f − ai have multiplicity at leastmi , i = 1, . . . , q. Then

q∑

i=1

(

1 − 1

mi

)

≤ 2.

3 The Proofs

We first need the following Lemmas:

Lemma 3.1 [13] Let xd−qii Di (x1, x2, . . . , xN+1),1 ≤ i ≤ s, be homogeneous poly-

nomials of degree d which determine s hypersurfaces in general position of PN (C).

Suppose there exists a holomorphic curve f from C to PN (C) with the reduced rep-

resentation f̃ = ( f1 : · · · : fN+1) be such that its image lies in the curve definedby

s∑

i=1

xd−qii Di (x1, x2, . . . , xN+1) = 0, d ≥ s2 +

s∑

i=1

qi .

Then the polynomials xd−q11 D1(x1, x2, . . . , xN+1), . . . , xd−qs

s Ds(x1, x2, . . . , xN+1)

are linearly dependent on the image of f .

Lemma 3.2 [16] Let f1, . . . , fn+1 be non-zero holomorphic functions on C, satisfyingf d1 + f d

2 + · · · + f dn+1 = 0. Assume that d ≥ n2.Then there is a decomposition of

indices, {i = 1, . . . , n + 1} = ∪Iv, such that

1. Every Iv contains at least 2 indices;2. fi = ci j f j , ci j is a non-zero constant for j, i ∈ Iv for all v;3.

∑j∈Iv f d

j = 0 for all v.

Lemma 3.3 Let a1, a2, . . . , aq be distinct points of C, n, n1, n2, . . . , nq ∈ N∗, c ∈ C,

c �= 0 and q > 2 + ∑qi=1

nin . Then the functional equation

( f − a1)n1( f − a2)

n2 . . . ( f − aq)nq = cgn

has no non-constant meromorphic solutions ( f, g).

Proof Suppose that( f, g) is a pair of non-constant meromorphic solutions of the equa-tion:

( f − a1)n1( f − a2)

n2 . . . ( f − aq)nq = cgn .

H. H. Khoai et al.

If z0 ∈ C is a zero of f − ai for some 1 ≤ i ≤ q, then z0 is a zero of g andniν f −ai (z0) = nνg(z0). So ν f −ai (z0) = n

niνg(z0) ≥ n

ni. Therefore N1, f (ai , r) ≤

nin N f (ai , r) ≤ ni

n T f (r) + S f (r). From this and by Lemma 2.1,

(q − 2)T f (r) ≤q∑

i=1

N1, f (ai , r) + S f (r) ≤q∑

i=1

ni

nN f (ai , r) + S f (r)

≤q∑

i=1

ni

nT f (r) + S f (r),

(

q − 2 −q∑

i=1

ni

n

)

T f (r) ≤ S f (r).

Therefore we get q ≤ 2 + ∑qi=1

nin , a contradiction.

Now consider polynomial P(z) = zn −azn−m +b, where a, b ∈ C and n, m ∈ N∗,

satisfying the following condition:

a �= 0, b �= 0; n ≥ 2m + 9; m ≥ 2; (m, n) = 1. (3.1)

Lemma 3.4 Assume that P(z) satisfies the condition (3.1), and let f, g be two non-constant meromorphic functions. If P( f ) = P(g), then f ≡ g.

Proof The proof of Lemma 3.4 follows immediately from [6].

Lemma 3.5 Assume that P(z) satisfies the condition (3.1), and let f = f1

f2, g = g1

g2be two non-constant meromorphic functions. Suppose that

f n1 − a f n−m

1 f m2 + b f n

2 = gn1 − agn−m

1 gm2 + bgn

2 .

Then g1 = c f1, g2 = c f2, with cn = 1.

Proof By the hypothesis,

b f n2 + f n−m

1 ( f m1 − a f m

2 ) − bgn2 − gn−m

1 (gm1 − agm

2 ) = 0. (3.2)

Note that each common zero of

f n2 , f n−m

1 ( f m1 − a f m

2 ), gn2 , gn−m

1 (gm1 − agm

2 )

is also a common zero of f1, f2, g1, g2, so we can eliminate these common zeros.Therefore, without loss of generality, we may assume that

F1 = ( f2, f1, g2, g1)

defines a holomorphic curve from C to P3(C).


Moreover bxn1 , xn−m

2 (xm2 − axm

1 ), bxn3 , xn−m

4 (xm4 − axm

3 ) are the homogeneouspolynomials of degree n which determine hypersurfaces in general position of P

3(C).

Since n ≥ 2m+9 and by Lemma 3.1, there exist constants C1, C2, C3, (C1, C2, C3) �=(0, 0, 0), such that

C1b f n2 + C2 f n−m

1 ( f m1 − a f m

2 ) + C3bgn2 = 0. (3.3)

We consider the following possible cases:

Case 1: C3 = 0. Then from (3.3) we have

C1b f n2 + C2 f n−m

1 ( f m1 − a f m

2 ) = 0. (3.4)

It is easy to see that f is constant.So C3 �= 0.Case 2: C2 = 0. From (3.3) we have

C1b f n2 + C3bgn

2 = 0.

Then we have C1 �= 0, C3 �= 0. From this and (3.2) it follows that

gn2 = −C1b

C3bf n2 ,

g2

f2= c, c �= 0,

b(

1 + C1

C3

)f n2 + f n−m

1 ( f m1 − a f m

2 ) − gn−m1 (gm

1 − acm f m2 ) = 0. (3.5)

Suppose that 1 + C1

C3�= 0. Then, from the similarity of (3.5) and (3.2), by the similar

arguments we see that there exist the constants C ′1, C ′

2, (C ′1, C ′

2) �= (0, 0), such that

C ′2 f n

2 + C ′1 f n−m

1 ( f m1 − a f m

2 ) = 0. (3.6)

As above we obtain a contradiction. So 1 + C1

C3= 0. Therefore bgn

2 = b f n2 .

Case 3. C1 = 0. From (3.3) we have

C2 f n−m1 ( f m

1 − a f m2 ) + C3bgn

2 = 0.

It is easy to see that C2 �= 0, C3 �= 0. We have

C2 f n−m1 ( f m

1 − a f m2 ) = −C3bgn

2 , C2

( f1

f2

)n − C2a( f1

f2

)n−m = −C3b( g2

f2

)n.

(3.7)

H. H. Khoai et al.

Note that the equation zm − a = 0 has m distinct roots a1, a2, . . . , am . Set f =f1f2

, g = g2f2

. From (3.7) and this it follows that

f n−m( f − a1) · · · ( f − am) = cgn, c �= 0. (3.8)

Sincef1

f2is non-constant, so is

g2

f2, too. From (3.8) and by (m, n) = 1 we see that the

multiplicity of each zero of f and f − ai is a multiple of n. By n ≥ 2m + 9, m ≥ 2and Lemma 2.2 we conclude that the Eq. (3.8) has no non-constant meromorphicsolutions.

Case 4. C1 �= 0, C2 �= 0, C3 �= 0.By the similar arguments as in (3.5) we obtain a contradiction. So b f n

2 = bgn2 , f n

2 =gn

2 , and g2 = c f2 with cn = 1.

Now we can return to the proof of Lemma 3.5. By

f n1 − a f n−m

1 f m2 + b f n

2 = gn1 − agn−m

1 gm2 + bgn

2 ,

and f n2 = gn

2 , and by Lemma 3.4, we obtainf1

f2= g1

g2. Thus

g1

f1= g2

f2. Therefore

g1 = c f1, g2 = c f2 with cn = 1.

Lemma 3.6 Let f = f1

f2, g = g1

g2be two non-constant meromorphic functions and

let n, m ∈ N∗, n ≥ 2m + 9, a1, b1, c1, a2, b2, c2 ∈ C, a1, b1, c1, a2, b2, c2 �= 0.

Suppose that

a1 f n1 + b1 f n−m

1 f m2 + c1 f n

2 = a2gn1 + b2gn−m

1 gm2 + c2gn

2 . (3.9)

If either m ≥ 2, (m, n) = 1, or m ≥ 4, then g1 = c f1, g2 = d f2 with a1 = a2cn, b1 =b2cn−mdm, c1 = c2dn .

Proof By the similar arguments as in the proof of Lemma 3.5 we obtain c1 f n2 = c2gn

2 .Therefore g2 = d f2 with c1 = c2dn . From this and (3.9) we have

c1 f n2

(a1

c1f n + b1

c1f n−m + 1

)

= c2gn2

(a2

c2gn + b2

c2gn−m + 1

)

,

Seta1

c1= a3,

b1

c1= b3,

a2

c2= a4,

b2

c2= b4. Then

a3 f n + b3 f n−m = a4gn + b4gn−m . (3.10)

Set h = g

f. From this and (3.9) we obtain

a3 f m + b3 = a4

(g

f

)n

f m + b4

(g

f

)n−m

, a3 f m + b3 = a4hn f m + b4hn−m,


f m(a3 − a4hn) = b4hn−m − b3,−a4

(

hn − a3

a4

)

b4

(

hn−m − b3

b4

) =(

1

f

)m

. (3.11)

Assume that h is not constant. Consider the following possible cases:

Case 1. m ≥ 2, (m, n) = 1. If hn − a3

a4= 0, hn−m − b3

b4= 0 have no common

zeros, then all zeros of hn − a3

a4have multipcities ≥ m. Then

N1,hn

(a3

a4, r

)

≤ 1

mNhn

(a3

a4, r

)

.

By Lemmas 2.1 we obtain

Thn (r) ≤ N1,hn (∞, r) + N1,hn (0, r) + N1,hn

(a3

a4, r

)

+ S f (r),

nTh(r) ≤ 2Th(r) + n

mNh

(a3

a4, r

)

+ S f (r) ≤(

2 + n

m

)Th(r) + S f (r)

(n − 2 − n

m

)Th(r) ≤ S f (r),

which leads to n(m − 1) ≤ 2m, a contradiction to the condition n ≥ 2m + 9. If

hn − a3

a4= 0, hn−m − b3

b4= 0 have common zeros, then there exists z such that

hn(z) = a3

a4, hn−m(z) = b3

b4. Set a = h(z). From (3.11) we get

− a4(hn − an)

b4(hn−m − an−m)=

(1

f

)m

,−a4an

((h

a

)n

− 1

)

b4an−m

((h

a

)n−m

− 1

)

=(

1

f

)m

,−a4am(hn1 − 1)

b4(hn−m1 − 1)

=(

1

f

)m

, (3.12)

where h1 = h

a. Since (m, n) = 1, the equations zn − 1 = 0 and zn−m − 1 = 0 have

different roots except for z = 1.

Let ri , i = 1, . . . , 2n − m − 2 be all the roots of them. Then all zeros of h − ri

have multipcities ≥ m. Therefore, by Lemma 2.2 we obtain

(

1 − 1

m

)

(2n − m − 2) ≤ 2, 2n(m − 1) ≤ m2 + 3m − 2,

which contradicts n ≥ 2m + 9. Thus h is a constant and so is g1 = c f1 , too.

H. H. Khoai et al.

Case 2. m ≥ 4. Note that equation zn − a3

a4= 0 has n simple zeros, equation

zn−m − b3

b4= 0 has n − m simple zeros. Then zn − a3

a4= 0, zn−m − b3

b4= 0 have at

most n − m common simple zeros.Therefore the equation zn − a3

a4= 0 has at least m

distinct roots, which are not roots of zn−m − b3

b4= 0. Let r1, r2, . . . , rm be all these

roots. Then all the simple zeros of h − r j , j = 1, . . . , m, have multipcities ≥ m. By

Lemma 2.2 we have m(1 − 1

m) ≤ 2. Therefore m ≤ 3. From m ≥ 4, we obtain a

contradiction. Thus h is a constant and so is g1 = c f1, too. So g1 = c f1, g2 = d f2.

From (3.11) and Lemma 3.3, and since f, g are non-constant fumctions we obtaina1 = a2cn, b1 = b2cn−mdm, c1 = c2dn . Lemma 3.6 is proved.

Lemma 3.7 Let f, g be two non-degenerate holomorphic mappings from C to PN (C),

and let T (z1, z1, . . . , zN+1) defined as in Theorem 1. If T ( f̃ ) = T (g̃), then f ≡ g.

Proof It is easy to see that T di ( f̃ ) �≡ 0, i = 1, 2, . . . , q − 1. Indeed, if there is

i ∈ {1, . . . , q − 1} such that T di ( f̃ ) ≡ 0, then

Lni+1( f̃ ) − ai Ln−m

i+1 ( f̃ )Lm1 ( f̃ ) + bi Ln

1( f̃ ) ≡ 0. (3.13)

Therefore,Li+1( f̃ )

L1( f̃ )is a constant, and we have a contradiction to the linearly

degeneracy of f. Similarly, T di (g̃) �≡ 0, i = 1, 2, . . . , q − 1.

Consider the equation

T ( f̃ ) − T (g̃) = 0. (3.14)

Since d ≥ (2q − 3)2, from Lemmas 3.2 and 3.6 it follows that for each i =1, 2, . . . , q − 1, there exists an unique j ∈ {1, . . . , q − 1} such that


i+1 ( f̃ )Lm1 ( f̃ ) + bi Ln

1( f̃ ) = ci j (Lnj+1(g̃) − a j Ln−m

j+1 (g̃)Lm1 (g̃)

+b j Ln1(g̃)), (3.15)

where cdi j = 1.

Now we are going to show that i = j . Assume, on the contrary, that i �= j . Thenthere exist l such that i �= l,


i+1 ( f̃ )Lm1 ( f̃ ) + bi Ln

1( f̃ ) = ci j (Lnj+1(g̃) − a j Ln−m

j+1 (g̃)Lm1 (g̃)

+b j Ln1(g̃)),

Lnl+1( f̃ ) − al Ln−m

l+1 ( f̃ )Lm1 ( f̃ ) + bl Ln

1( f̃ ) = cli (Lni+1(g̃) − ai Ln−m

i+1 (g̃)Lm1 (g̃)

+bi Ln1(g̃)).


From Lemma 3.6 we obtain

bi Ln1( f̃ ) = ci j b j Ln

1(g̃), bl Ln1( f̃ ) = cli bi Ln

1(g̃).

So

bdi (Ln

1( f̃ ))d = cdi j b

dj (Ln

1(g̃))d , bdl (Ln

1( f̃ ))d = cdli b

di (Ln

1(g̃))d .

Because cdi j = 1 and cd

li = 1,

b2di = bd

j bdl .

We obtain a contradiction. Thus i = j, and


i+1 ( f̃ )Lm1 ( f̃ ) + bi Ln

1( f̃ ) = cii (Lni+1(g̃) − ai Ln−m

i+1 (g̃)Lm1 (g̃)

+bi Ln1(g̃)),

for all i = 1, 2, . . . , q −1. By the similar arguments as in Lemma 3.5, we get Ln1( f̃ ) =

cii Ln1(g̃). Therefore

Ln1( f̃ )

((Li+1( f̃ )

L1( f̃ )

)n

− ai

(Li+1( f̃ )

L1( f̃ )

)n−m

+ bi

)

= cii Ln1(g̃)

((Li+1(g̃)

L1(g̃)

)n

− ai

(Li+1(g̃)

L1(g̃)

)n−m

+ bi

)

,

(Li+1( f̃ )

L1( f̃ )

)n

− ai

(Li+1( f̃ )

L1( f̃ )

)n−m

+bi =(

Li+1(g̃)

L1(g̃)

)n

− ai

(Li+1(g̃)

L1(g̃)

)n−m

+bi ,

for all i = 1, 2, . . . , q − 1. By Lemma 3.4 we have

Li+1( f̃ )

L1( f̃ )= Li+1(g̃)

L1(g̃), i = 1, 2, . . . , q − 1.

So

L1( f̃ )

L1(g̃)= L2( f̃ )

L2(g̃)= · · · = Ln+1( f̃ )

Ln+1(g̃)= · · · = Lq( f̃ )

Lq(g̃)= c.

Therefore

Li ( f̃ ) = cLi (g̃), i = 1, 2, . . . , n + 1, . . . , q.

H. H. Khoai et al.

From this we obtain

f1

g1= f2

g2= · · · = fN+1

gN+1.

Thus f ≡ g.Now we are going to complete the proof of Theorem 1.Let f̃ = ( f1 : · · · : fN+1) and g̃ = (g1 : · · · : gN+1) be reduced representations

of f and g, respectively.Since E f (X) = Eg(X), it is easy to see that there exists an entire function c such

that T ( f̃ ) = ecT (g̃). Set l = ec

nd and h̃ = (lg1 : · · · : lgN+1). Then h̃ is a reducedrepresentation of g and T ( f̃ ) = T (h̃). By Lemma 3.7, f ≡ g.

As a connection to the study of the uniqueness problem, many authors (see [1,6,12,15,17–21]) introduced the following definition:

Definition 3.1 A polynomial P ∈ C[z] is called an uniqueness polynomial (UPM forshort) for meromorphic functions if P( f ) = P(g) then f = g for all non-constantmeromorphic functions f, g on C.

If instead of meromorphic functions we consider holomorphic curves, then we canintroduce the following definition.

Definition 3.2 A homogeneous polynomial P of variables z1, . . . , zN+1 is said tobe an uniqueness polynomial for holomorphic curves if the condition P( f̃ ) = P(g̃)

implies f = g for all linearly non-degenerate holomorphic curves f and g from C toP

N (C) with reduced representations f̃ and g̃, respectively.

Then Lemma 3.7 can be reformulated as follows.

Theorem 3.8 The polynomial T (z1, z2, . . . , zN+1) is an uniqueness polynomial forholomorphic curves.

Acknowledgments The authors would like to thank the referee for his/her valuable suggestions.

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uniqueness theorems for holomorphic curves with hypersurfaces of fermat–waring type

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