C*-algeb aic Popo iciu Conjec u e
K. MAHESH KRISHNA
School o Ma hema ics and Na u al Sciences
Chanakya Uni e si y Global Campus
NH-648, Ha alu u Village
De anahalli Taluk, Bengalu u Ru al Dis ic
Ka na aka S a e 562 110 India
Email: [email p o ec ed]
Da e: No embe 24, 2025
Abs ac : We o mula e C*-algeb aic e sion o Popo iciu Conjec u e (Rahman-Sudbe y Theo em) and
show ha i holds o deg ee 2 polynomials o e commu a i e uni al C*-algeb as.
Keywo ds: Popo iciu Conjec u e, C*-algeb a.
Ma hema ics Subjec Classi ica ion (2020): 30C15, 46L05.
Le C[z] be he se o all polynomials o e C. Conjec u e o Popo iciu [2] which is la e p o ed pa ially
by Rahman [1] and ully by Sudbe y [3] s a es he ollowing.
Theo em 0.1. [1–3] (Popo iciu Conjec u e/Rahman-Sudbe y Theo em) Le n≥2and p(z) =
(z−a1)(z−a2)· · · (z−an)∈C[z]be such ha aj6=ak o some 1≤j, k ≤n, j 6=k. Then he polynomial
q(z):=p(z)p0(z)· · · p(n−1)(z)∈C[z]
has a leas n+ 1 dis inc ze os.
In his no e, we o mula e C*-algeb aic analogue o Theo em 0.1 and e i y i o deg ee 2 polynomials
o e commu a i e uni al C*-algeb as.
Conjec u e 0.2. (C*-algeb aic Popo iciu Conjec u e) Le Abe a uni al commu a i e C*-algeb a.
Le n≥2and p(z)=(z−a1)(z−a2)· · · (z−an)∈ A[z]be such ha aj6=ak o some 1≤j, k ≤n, j 6=k.
Then he polynomial
q(z):=p(z)p0(z)· · · p(n−1)(z)∈ A[z]
has a leas n+ 1 dis inc ze os.
Theo em 0.3. Conjec u e 0.2 holds o deg ee 2 polynomials.
P oo . Le Abe a uni al commu a i e C*-algeb a. Le
p(z) = (z−a)(z−b)∈ A[z]
wi h a6=b. Then
q(z) = p(z)p0(z)=(z−a)(z−b)(2z−(a+b)).
Hence
p(a) = p(b) = pa+b
2= 0.
1
K. MAHESH KRISHNA
No e ha
a6=b, a 6=a+b
2, b 6=a+b
2.
Re e ences
[1] Q. I. Rahman. The dis inc ze os o he p oduc o a polynomial and i s successi e de i a i es. Can. Ma h. Bull.,
14:267–269, 1971.
[2] Q. I. Rahman and G. Schmeisse . Analy ic heo y o polynomials, olume 26 o Lond. Ma h. Soc. Monog ., New Se .
Ox o d: Ox o d Uni e si y P ess, 2002.
[3] A. Sudbe y. The numbe o dis inc oo s o a polynomial and i s de i a i es. Bull. Lond. Ma h. Soc., 5:13–17, 1973.
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