A PROOF OF SENDOV’S CONJECTURE T. AGAMA Abstract. In this paper we give a proof of Sendov’s conjecture. We start by establishing the uniformly diminishing state of the mass of an expansion. 1. Introduction The sendov conjecture is the assertion that any complex coefficient polynomial Pn (z) of degree n ≥ 2 with sufficiently small zeros must sit in the same unit disc with at least one of its critical point. That is is to say, for each |bi | < 1 such that Pn (bi ) = 0, then there exist some cj such that |bi − cj | < 1 where Pn′ (cj ) = 0. There has been various successful attacks on variants of the conjecture most of which proceeded by the methods of complex variable and classical analysis, which is not surprising given the origin of the problem. Though it seems the state-of-art approach to the problem might not guarantee a solution, the results are noteworthy. In [1] the conjecture has been proved for polynomials of degree at most six. This was improved to polynomials of degree at most seven in [2] and polynomials of degree at most eight in [4]. An asymptotic version of the conjecture was also recently shown to hold [3]. In this paper, however, we adopt and follow an unconventional approach in resolving this conjecture. We start by developing some basic tools that allows us to obtain a general and a much stronger version of Sendov’s conjecture. Consequently, we managed to prove the result Theorem 1.1. Let f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 ∈ C[x]. Let T = {b1 , b2 , . . . , bn } and C = {c1 , c2 , . . . , cn−1 } be the set of zeros and critical values of f , respectively. If |bi | < δ ≤ 1, for i = 1, 2, . . . n, then for each bi ∈ T , there exist some cj ∈ C such that |bi − cj | < 1. We also generalized this result which allows us to say something about the distribution of the zeros of any polynomial and the zeros of its higher order derivatives, as follows: Date: October 21, 2019. 2000 Mathematics Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20. Key words and phrases. sub-expansion; regularity; momentum; mass; boundary points; index. 1 2 T. AGAMA Theorem 1.2. Let f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 ∈ C[x]. Let T = {b1 , b2 , . . . , bn } be the set of zeros of f . If |bi | < δ ≤ 1 for i = 1, 2, . . . n, then for each bi ∈ T , there exist some cj with f n (cj ) = 0 such that |bi − cj | < 1 for all 1 ≤ n ≤ deg(f ) − 1. 2. Notations Through out this paper a tuple will always be denoted by S or Sj where j ∈ N. Occasionally, we will use the tuple SC to denote a tuple of the base field and SC[x] a tuple of the polynomial ring C[x]. We set S0 = (0, 0, . . . , 0) and call it the null tuple and Se = (1, 1, . . . , 1) the unit tuple. We denote the rank of an expansion on S by R(S), the degree of an expansion on S by deg(S) and the measure of an expansion on S by N (S). Also we set Sa = (f1 (a), f2 (a), . . . , fn (a)), where S = (f1 , f2 , . . . , fn ). 3. Preliminary definitions and terminologies In this section we introduce the following language. Definition 3.1. Let S = (f1 , f2 , . . . , fn ) such that each fi ∈ C[x]. By the derivative of S denoted ∇(S), we mean   dfn df1 df2 . , ,..., ∇(S) = dx dx dx We denote the derivative of this tuple at a point a ∈ R to be   dfn (a) df1 (a) df2 (a) . , ,..., ∇a (S) = dx dx dx Definition 3.2. Let {Si }∞ i=1 be a collection of tuples of C[x]. Then by an expansion on {Si }∞ , we mean the composite map i=1 ∞ γ −1 ◦ β ◦ γ ◦ ∇ : {Si }∞ i=1 −→ {Si }i=1 , where   f1  f2    γ(S) =  .   ..  fn and  0 1 1 0  β(γ(S)) =  . .  .. .. 1 1 ··· ··· ··· ···   1 f1  f2  1   ..   ..  . .  .  0 fn ∞ Proposition 3.1. An expansion γ −1 ◦ β ◦ γ ◦ ∇ : {Si }∞ i=1 −→ {Si }i=1 is linear. 4. The rank and measures of an expansion In this section we introduce the notion of the rank and measure of an expansion. We launch the following languages as follows: Definition 4.1. Let F = {Sm }∞ m=1 be collection of tuples of C[x]. Then the value of n such that the expansion (γ −1 ◦β◦γ◦∇)n (S) 6= S0 and (γ −1 ◦β◦γ◦∇)n+1 (S) = S0 is called the degree of expansion and (γ −1 ◦ β ◦ γ ◦ ∇)n (S) is called the rank of an expansion, denoted R(S). A PROOF OF SENDOV’S CONJECTURE 3 Theorem 4.2. Let Si , Sj ∈ {Sk }∞ k=1 , a family of tuples of C[x]. Let deg(Si ) = deg(Sj ). Then R(Si ) = R(Sj ) if and only if Si − Sj = (a1 , a2 , . . . , an ) for each ai ∈ C. Proof. Pick Si , Sj ∈ {Sk }∞ k=1 such that deg(Si ) = deg(Sj ). Suppose Si − Sj = (a1 , a2 , . . . , an ), then by applying the nth expansion on both sides, we find that (γ −1 ◦ β ◦ γ ◦ ∇)n (Si − Sj ) = S0 . Since an expansion is linear, it follows that (γ −1 ◦ β ◦ γ ◦ ∇)n (Si ) − (γ −1 ◦ β ◦ γ ◦ ∇)n (Sj ) = S0 . That is R(Si ) = R(Sj ) + S0 = R(Sj ). Conversely, suppose R(Si ) = R(Sj ), then it follows that R(Si − Sj ) = S0 . To avoid a contradiction, we must allow the entries of Si and Sj to differ by elements of C. This completes the proof.  Definition 4.3. Let S be a tuple of C[x], then by the measure of an expansion on S, denoted N (S), we mean N (S) = ||R(S)||. 5. The boundary points and mass of an expansion In this section, we introduce the notion of the mass and boundary of an expansion on tuples of C[x]. Definition 5.1. Let {Sj }∞ j=1 be a collection of tuples of C[x]. By the boundary points of the nth expansion, denoted Z[(γ −1 ◦ β ◦ γ ◦ ∇)n (Sj )], we mean the set  Z[(γ −1 ◦ β ◦ γ ◦ ∇)n (Sj )] := (a1 , a2 , . . . , an ) : Idi [(γ −1 ◦ β ◦ γ ◦ ∇)nai (Sj )] = 0 . Remark 5.2. To avoid writing the boundary of expansion in the form Z[(γ −1 ◦ β ◦ γ ◦ ∇)n (Sj )], we choose to rather write B n (Sj ). Also for the nth expansion on Sj , we will choose to write Sjn . We will switch between these two notations occasionally without commenting too much about it. Definition 5.3. Let F = {Sj }∞ j=1 be a collection of tuples of C[x]. Then by the mass of an expansion Sjn , denoted H(Sjn ), we mean the finite sum X H(Sjn ) = ||Sk ||, Sk ∈Bn (Sj ) where v u n uX |ai |2 . ||Sk || = t i=1 for Sk = (a1 , a2 , . . . , an ) 6. The speed, momentum, index and embedding of an expansions In this section we launch the notion of the speed and momentum of an expansion. Definition 6.1. Let {Sj }∞ j=1 be a collection of tuples of C[x]. Then by the speed of an expansion on S, denoted ν(S), we mean the expression ν(S) = N (S) . deg(S) 4 T. AGAMA Definition 6.2. Let {Sj }∞ j=1 be a collection of tuples of C[x]. By the momentum of the nth expansion, denoted M(Sjn ), we mean M(Sjn ) := ν(Sjn )H(Sjn ). Definition 6.3. Let P = {Sj }nj=1 be a finite collection of tuples of C[x]. Then by the index of the mth expansion of Sk for 1 ≤ k ≤ n, denoted I(Skm ), we mean I(Skm ) = n P j=1 M(Sjm ) M(Skm ) . Definition 6.4. Let F = {Sj }∞ j=1 be a collection of tuples of C[x]. Let Sa , Sb ∈ F, −1 then we say the expansion (γ ◦ β ◦ γ ◦ ∇)n1 (Sb ) is an embedding of the expansion (γ −1 ◦ β ◦ γ ◦ ∇)n2 (Sa ) if (6.1) Z[(γ −1 ◦ β ◦ γ ◦ ∇)n1 (Sb )] ⊂ Z[(γ −1 ◦ β ◦ γ ◦ ∇)n2 (Sa )] for some n1 > n2 . Conversely, we say (γ −1 ◦ βγ ◦ ∇)n2 (Sa ) is an extension of the expansion (γ −1 ◦ β ◦ γ ◦ ∇)n1 (Sb ). Proposition 6.1. Let F = {Sj }∞ j=1 be a collection of n tuples of C[x], and suppose Sa , Sb ∈ F. If (γ −1 ◦ β ◦ γ ◦ ∇)n2 (Sa ) is an embedding of the expansion (γ −1 ◦ β ◦ γ ◦ ∇)n1 (Sb ), then H(San2 ) < H(Sbn1 ). Proof. Let Sa , Sb ∈ F and suppose (γ −1 ◦ β ◦ γ ◦ ∇)n2 (Sa ) is an embedding of the expansion (γ −1 ◦ β ◦ γ ◦ ∇)n1 (Sb ), then it follows from definition 6.4 Z[(γ −1 ◦ β ◦ γ ◦ ∇)n1 (Sb )] ⊂ Z[(γ −1 ◦ β ◦ γ ◦ ∇)n2 (Sa )] for some n1 , n2 ∈ N. The result follows from this fact by leveraging definition 5.3.  Remark 6.5. Next we establish an important inequality that relates the index of an expansion of a tuple to the largest size of the number of embedding of expansion, in the following result. Lemma 6.6. Let P := {Sj }nj=1 and suppose (γ −1 ◦ β ◦ γ ◦ ∇)nk (Sk ) (1 ≤ k ≤ n) n admits an embedding (γ −1 ◦ β ◦ γ ◦ ∇)nj (Sj ) for all 1 ≤ j ≤ n. If ν(Sknk ) ≥ ν(Sj j ) for all 1 ≤ j ≤ n, then I(Sknk ) < n. A PROOF OF SENDOV’S CONJECTURE 5 Proof. Let P := {Sj }nj=1 and suppose (γ −1 ◦ β ◦ γ ◦ ∇)nk (Sk ) (1 ≤ k ≤ n) admits an embedding (γ −1 ◦ β ◦ γ ◦ ∇)nj (Sj ) for all 1 ≤ j ≤ n. Then it follows that n X n M(Sj j ) = M(S1n1 ) + · · · + M(Sknk ) + · · · + M(Snnn ) j=1 = ν(S1n1 )H(S1n1 ) + · · · + ν(Sknk )H(Sknk ) + · · · + ν(Snnn )H(Snnn ) ≤ nν(Sknk )H(Sknk ) = nM(Sknk ), and the inequality is established.  Theorem 6.7. Let F = {Sj }∞ j=1 be a collection of tuples of C[x]. Then for each S∈F deg(S)−1 X ν(S k ) = ν(S)deg(S) log(deg(S)) + deg(S)ν(S)α + O(ν(S)), k=0 where α = 0.5772 · · · , the euler-macheroni constant. Proof. Clearly deg(S)−1 X ν(S k ) = ν(S) + ν(S 1 ) + · · · + ν(S deg(S)−1 ) k=0 N (S) N (S 1 ) N (S deg(S)−1 ) + + · · · + deg(S) deg(S 1 ) deg(S deg(S)−1 )   1 1 1 = N (S) + + ··· + deg(S) deg(S 1 ) deg(S deg(S)−1 )   1 1 1 = N (S) + + ··· + + 1 deg(S) deg(S) − 1 2 = deg(S) = ν(S)deg(S) X 1 m m=1 thereby establishing the formula.  Remark 6.8. This formula, as it turns out, becomes extremely useful in establishing the diminishing state of the mass of an expansion. For the time being, we use this formula to prove that the mass of an expansion diminishes at some phase. Theorem 6.9. Let F = {Sj }∞ j=1 be a collection of tuples of C[x]. Suppose S ∈ F, then H(S n ) > H(S n+1 ) for some 0 ≤ n ≤ deg(S) − 2. Proof. Let F = {Sj }∞ j=1 be a collection of tuples of C[x] and specify S ∈ F. deg(S)−1 Consider the finite collection P = {S k }k=0 . Suppose on the contrary that H(S n ) ≤ H(S n+1 ) 6 T. AGAMA for all 0 ≤ n ≤ deg(S) − 2. Then it follows by an application of Theorem 6.7 that deg(S)−1 X k=0 deg(S)−1 k M(S ) = X H(S k )ν(S k ) k=0 deg(S)−1 ≤ H(S deg(S)−1 ) X ν(S k ) k=0 ≪ H(S deg(S)−1 )ν(S)deg(S) log(deg(S)) ≤ H(S deg(S)−1 )ν(S deg(S)−1 )deg(S) log(deg(S)) = M(S deg(S)−1 )deg(S) log(deg(S)),   deg(S)−2 1 and it follows that the index of expansion I (S )) ≪ deg(S) log(deg(S)), thereby contradicting the upper bound in Lemma 6.6.  Next we conjecture a stronger version of Sendov’s conjecture as follows: Conjecture 6.1. Let f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 ∈ C[x] and let S = (f (x), f (x), · · · , f (x)), where f (x) has no repeated zeros. Suppose H(S) < 1, then for each S0 ∈ Z[(γ −1 ◦ β ◦ γ ◦ ∇)0 (S)] ||S0 − Sj || < 1 for all Sj ∈ Z[(γ −1 ◦ β ◦ γ ◦ ∇)m (S)] for all 1 ≤ m ≤ deg(S) − 1. 7. Regular and sub-expansions In this section we introduce the notion of regularity of an expansion and subexpansion of an expansion. Definition 7.1. Let F = {Sj }∞ j=1 be a collection of tuples of C[x]. Then for any Sk ∈ F, we say the expansion (γ −1 ◦ β ◦ γ ◦ ∇)n (Sk ) is regular if H(Skn ) > H(Skn+1 ) for some 0 ≤ n ≤ deg(Sk ) − 2. Theorem 7.2. Let F = {Sj }∞ j=1 be collection of tuples of C[x] and suppose the expansion (γ −1 ◦ β ◦ γ ◦ ∇)n (Sk ) with (n ≤ deg(Sk ) − 3) is regular for Sk ∈ F. If H(Skn ) < δ for 0 < δ < 1 sufficiently small, then for each Sl ∈ Z[(γ −1 ◦ β ◦ γ ◦ ∇)n (Sk )], there exist some S0 ∈ Z[(γ −1 ◦ β ◦ γ ◦ ∇)n+1 (Sk )] such that ||S1 − S0 || < 1. Proof. Let F = {Sj }∞ j=1 be collection of tuples of C[x]. Pick arbitrarily Sk ∈ F and suppose the expansion (γ −1 ◦ β ◦ γ ◦ ∇)n (Sk ) with (n ≤ deg(Sk ) − 3) is regular. Suppose on the contrary that for each Sl ∈ Z[(γ −1 ◦ β ◦ γ ◦ ∇)n (Sk )], then ||S1 − S0 || ≥ 1 A PROOF OF SENDOV’S CONJECTURE 7 for all S0 ∈ Z[(γ −1 ◦ β ◦ γ ◦ ∇)n+1 (Sk )]. Since H(Skn ) < δ with 0 < δ < 1 sufficiently small and n ≤ deg(Sk ) − 3, it follows that H(Skn+1 ) ≥ 1. Under the regularity condition, it must be that 1 > δ > H(Skn ) > H(Skn+1 ) ≥ 1 which is absurd . This completes the proof of the theorem.  Definition 7.3. Let (γ −1 ◦ β ◦ γ ◦ ∇)m (Sa ) and (γ −1 ◦ β ◦ γ ◦ ∇)n (Sb ) be any two expansions with m < n, then we say (γ −1 ◦ β ◦ γ ◦ ∇)m (Sa ) is a sub-expansion of the expansion (γ −1 ◦ β ◦ γ ◦ ∇)n (Sb ), denoted (γ −1 ◦ β ◦ γ ◦ ∇)m (Sa ) ≤ (γ −1 ◦ β ◦ γ ◦ ∇)n (Sb ) if there exist some j ≥ 1 such that (γ −1 ◦ β ◦ γ ◦ ∇)m (Sa ) = (γ −1 ◦ β ◦ γ ◦ ∇)m+j (Sb ). We say the expansion is proper if m + j = n. We denote this proper expansion by (γ −1 ◦ β ◦ γ ◦ ∇)m (Sa ) < (γ −1 ◦ β ◦ γ ◦ ∇)n (Sb ). Remark 7.4. Next we prove a result that indicates that the regularity condition on an expansion can be localized as well as extended through expansions. Theorem 7.5. Let (γ −1 ◦ β ◦ γ ◦ ∇)m (Sa ) < (γ −1 ◦ β ◦ γ ◦ ∇)n (Sb ), a proper subexpansion. Then (γ −1 ◦β ◦γ ◦∇)m (Sa ) is regular if and only if (γ −1 ◦β ◦γ ◦∇)n (Sb ) is regular. Proof. Let (γ −1 ◦β ◦γ ◦∇)m (Sa ) < (γ −1 ◦β ◦γ ◦∇)n (Sb ), a proper sub-expansion and suppose (γ −1 ◦ β ◦ γ ◦ ∇)m (Sa ) is regular. Then it follows that H(Sam ) > H(Sam+1 ) for some 1 ≤ m ≤ deg(Sa ) − 2. Then by definition 7.3, It follows that there exist some j ≥ 1 such that we can write (γ −1 ◦ β ◦ γ ◦ ∇)m (Sa ) = (γ −1 ◦ β ◦ γ ◦ ∇)m+j (Sb ). Since the expansion is proper, It follows that m + j = n and we have (γ −1 ◦ β ◦ γ ◦ ∇)m (Sa ) = (γ −1 ◦ β ◦ γ ◦ ∇)n (Sb ). It follows that H(Sam ) = H(Sbn ). Since (γ −1 ◦ β ◦ γ ◦ ∇)m+1 (Sa ) = (γ −1 ◦ β ◦ γ ◦ ∇)n+1 (Sb ) the regularity condition of the expansion (γ −1 ◦ β ◦ γ ◦ ∇)n (Sb ) also follows. The converse on the other hand follows the same approach.  Proposition 7.1. Let F = {Sj }∞ j=1 be a collection of tuples of C[x]. Then the set  −1 G = (γ ◦ β ◦ γ ◦ ∇)n (S) : n ∈ N ∪ {0}, S ∈ F is a group. Proof. Clearly the set G is non-empty, since any tuple S ∈ F has a representation S = (γ −1 ◦ β ◦ γ ◦ ∇)0 (S). The null tuple S0 is the neutral element of the set. Pick (γ −1 ◦ β ◦ γ ◦ ∇)n (S) ∈ G, then it turns out that −(γ −1 ◦ β ◦ γ ◦ ∇)n (S) ∈ G is the inverse element, since an expansion is linear and for any tuple S ∈ F, then −S ∈ F. By the linearity of expansion, the set G satisfies the associative property. This proves that G is a group.  8 T. AGAMA Remark 7.6. Next we prove a result that indicates that the structure of an expansion is preserved at each phase of expansion. Theorem 7.7. Let  G = (γ −1 ◦ β ◦ γ ◦ ∇)n (S) : n ∈ N ∪ {0}, S ∈ F and then G ≃ G ′ .  G ′ = (γ −1 ◦ β ◦ γ ◦ ∇)n+1 (S) : n ∈ N ∪ {0}, S ∈ F , Proof. Consider the map λ : G −→ G ′ , where and with  G = (γ −1 ◦ β ◦ γ ◦ ∇)n (S) : n ∈ N ∪ {0}, S ∈ F  G ′ = (γ −1 ◦ β ◦ γ ◦ ∇)n+1 (S) : n ∈ N ∪ {0}, S ∈ F , λ[(γ −1 ◦ β ◦ γ ◦ ∇)n (S)] = (γ −1 ◦ β ◦ γ ◦ ∇)n+1 (S). We claim that the map is well-defined. For suppose S1 = S2 + SC with S1 6= S2 , then by appealing to Theorem 4.2 (γ −1 ◦ β ◦ γ ◦ ∇)n (S1 ) = (γ −1 ◦ β ◦ γ ◦ ∇)n (S2 + SC ) = (γ −1 ◦ β ◦ γ ◦ ∇)n (S2 ), and it follows that (γ −1 ◦ β ◦ γ ◦ ∇)n+1 (S1 ) = (γ −1 ◦ β ◦ γ ◦ ∇)n+1 (S2 ), by applying an extra copy of expansion on both sides. This proves that the map is independent on the choice of representative of tuples of C[x] in the same equivalence class. We claim that the map is injective. Suppose λ[(γ −1 ◦ β ◦ γ ◦ ∇)n (S1 )] = λ[(γ −1 ◦ β ◦ γ ◦ ∇)n (S2 )]. Then it follows that (γ −1 ◦ β ◦ γ ◦ ∇)n+1 (S1 ) = (γ −1 ◦ β ◦ γ ◦ ∇)n+1 (S2 ). We can remove one copy of expansion on both sides and still preserve unicity of both elements upto equivalence. Thus it follows that (γ −1 ◦ β ◦ γ ◦ ∇)n (S1 ) = (γ −1 ◦ β ◦ γ ◦ ∇)n (S2 ) + SC . This proves injectivity. Surjectivity follows by virtue of definition of the map. Finally we claim that the map λ so defined is a homomorphism. Consider the map (7.1) λ[(γ −1 ◦ β ◦ γ ◦ ∇)n (S1 ) + (γ −1 ◦ β ◦ γ ◦ ∇)n (S2 )]. Since expansion is linear, it follows that (7.1) = λ[(γ −1 ◦ β ◦ γ ◦ ∇)n (S1 )] + λ[(γ −1 ◦ β ◦ γ ◦ ∇)n (S2 )]. Thus the map is an isomorphism.  A PROOF OF SENDOV’S CONJECTURE 9 Theorem 7.8. Let S = (f1 (x), f2 (x), . . . , fn (x)) where each fi (x) has no repeated zeros. Then H(S m ) > H(S m+1 ) for all 0 ≤ m ≤ deg(S) − 2. Proof. Let S = (f1 (x), f2 (x), . . . , fn (x)) where each fi (x) has no repeated zeros. Then by appealing to Theorem 6.9 there exist some N0 ≥ 1 such that H(S N0 ) > H(S N0 +1 ). Then by applying Theorem 7.5 and Theorem 7.7, It follows that for all N ≥ N0 , then we have H(S N ) > H(S N +1 ). Again by appealing to Theorem 7.5 and Theorem 7.7, It follows that for all 0 < N ≤ N0 , then H(S N −1 ) > H(S N ). Combining these two cases, we obtain the following decreasing sequence of the mass of expansion H(S) = H(S 0 ) > H(S 1 ) > · · · > H(S deg(S)−3 ) > H(S deg(S)−2 ). This proves the diminishing state of the mass of an expansion.  Theorem 7.9. Let f (x) = an xn +an−1 xn−1 +· · ·+a1 x+a0 ∈ C[x] for n ≥ 3 and let S = (f (x), f (x), · · · , f (x)), where f (x) has no repeated zeros. Suppose H(S) < 1, then for each S0 ∈ Z[(γ −1 ◦ β ◦ γ ◦ ∇)0 (S)] ||S0 − Sj || < 1 for all Sj ∈ Z[(γ −1 ◦ β ◦ γ ◦ ∇)(S)]. Proof. Let f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 ∈ C[x] for n ≥ 3 and let S = (f (x), f (x), · · · , f (x)). Suppose H(S) < 1, then by applying Theorem 7.8, It follows that H(S 1 ) < H(S 0 ) = H(S) < 1. The result follows from this fact.  8. Proof of sendov conjecture We are now ready to prove Sendov’s conjecture. We assemble the tools we have developed thus far to solve the problem. We state our first theorem: Theorem 8.1. Let f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 ∈ C[x] with n ≥ 3. Let T = {b1 , b2 , . . . , bn } and C = {c1 , c2 , . . . , cn−1 } be the set of zeros and critical values of f , respectively. If |bi | < δ ≤ 1 for i = 1, 2, . . . n, then for each bi ∈ T , there exist some cj ∈ C such that |bi − cj | < 1. Proof. Let f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 ∈ C[x] with n ≥ 3 and let S = (f (x), f (x), · · · , f (x)). Let T = {b1 , b2 , . . . , bn } and C = {c1 , c2 , . . . , cn−1 } be the set of zeros and critical values of f , respectively, with |bi | < δ ≤ 1 for 10 T. AGAMA i = 1, 2, . . . n. Then we set H(S 0 ) < 1. Applying Theorem 7.9, It follows that for any (bτ (1) , bτ (2) , . . . , bτ (n) ) ∈ Z[(γ −1 ◦ β ◦ γ ◦ ∇)0 (S)], then it must be the case that (bτ (1) , bτ (2) , . . . , bτ (n) ) − (cα(1) , cα(2) , . . . , cα(n) ) < 1 for all (cα(1) , cα(2) , . . . , cα(n) ) ∈ Z[(γ −1 ◦ β ◦ γ ◦ ∇)(S)] where τ, α : {1, 2, . . . , n} −→ {1, 2, . . . n}. Since each entry of (bτ (1) , bτ (2) , . . . , bτ (n) ) is a zero of f (x) and each entry of (cα(1) , cα(2) , . . . , cα(n) ) is a critical value of f (x) for all permutations α, τ : {1, 2 . . . , n} −→ {1, 2, . . . , n}, the result follows immediately.  9. Extension of Sendov’s conjecture and further discussions It turns out that the method we have adopted in this paper can also be extended to not only the critical values of an arbitrary polynomial but as well to the zeros of a general class of polynomials of the form Pnm (x) for 1 ≤ m ≤ deg(Pn ) − 1. Since the mass of an expansion diminishes uniformly, we obtain a variant of Theorem 7.9: Theorem 9.1. Let f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 ∈ C[x] with n ≥ 3 and let S = (f (x), f (x), · · · , f (x)), where f (x) has no repeated zeros. Suppose H(S) < 1, then for each S0 ∈ Z[(γ −1 ◦ β ◦ γ ◦ ∇)0 (S)] ||S0 − Sj || < 1 for all Sj ∈ Z[(γ −1 m ◦ β ◦ γ ◦ ∇) (S)] for all 1 ≤ m ≤ deg(S) − 1. It follows from this result, an extension of the sendov conjecture: Theorem 9.2. Let f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 ∈ C[x] with n ≥ 3. Let T = {b1 , b2 , . . . , bn } be the set of zeros of f . If |bi | < δ ≤ 1 for i = 1, 2, . . . n, then for each bi ∈ T , there exist some cj with f n (cj ) = 0 such that |bi − cj | < 1 for all 1 ≤ n ≤ deg(f ) − 1. Even stronger than this is the assertion that Theorem 9.3. Let f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 ∈ C[x] and let S = (f (x), f (x), · · · , f (x)), where f (x) has no repeated zeros. Suppose H(S) < ǫ, then for each S0 ∈ Z[(γ −1 ◦ β ◦ γ ◦ ∇)0 (S)] ||S0 − Sj || < ǫ for all Sj ∈ Z[(γ −1 ◦ β ◦ γ ◦ ∇)m (S)] for all 1 ≤ m ≤ deg(S) − 1. The upshot of this is the result Theorem 9.4. Let f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 ∈ C[x] with n ≥ 3. Let T = {b1 , b2 , . . . , bn } be the set of zeros of f . If |bi | < δ ≤ ǫ for i = 1, 2, . . . n, then for each bi ∈ T , there exist some cj with f n (cj ) = 0 such that |bi − cj | < ǫ for all 1 ≤ n ≤ deg(f ) − 1. A PROOF OF SENDOV’S CONJECTURE 1 11 . References 1. Brown, Johnny E, On the Sendov conjecture for sixth degree polynomials,Proceedings of the American mathematical Society, vol. 175:4, 1991, pp 939–946. 2. Borcea, Lulius The Sendov conjecture for polynomials with at most seven distinct zeros, Analysis, vol. 16:2, Oldenbourg wissenschaftsverlag, 1996, pp 137–160. 3. Dégot, Jérôme, Sendov conjecture for high degree polynomials, Proceedings of the American Mathematical Society, vol. 142:4, 2014, pp 1337–1349. 4. Brown, Johnny E and Xiang, Guangping Proof of the Sendov conjecture for polynomials of degree at most eight, Journal of mathematical analysis and applications, vol. 232:2, Elsevier, 1999, pp 272–292. Department of Mathematics, African Institute for Mathematical science, Ghana E-mail address: theophilus@aims.edu.gh/emperordagama@yahoo.com 1 .