C*-algebraic Popoviciu Conjecture

C*-algebraic Popoviciu Conjecture K. MAHESH KRISHNA School of Mathematics and Natural Sciences Chanakya University Global Campus NH-648, Haraluru Village Devanahalli Taluk, Bengaluru Rural District Karnataka State 562 110 India Email: [email protected] Date: November 24, 2025 Abstract: We formulate C*-algebraic version of Popoviciu Conjecture (Rahman-Sudbery Theorem) and show that it holds for degree 2 polynomials over commutative unital C*-algebras. Keywords: Popoviciu Conjecture, C*-algebra. Mathematics Subject Classification (2020): 30C15, 46L05. Let C[z] be the set of all polynomials over C. Conjecture of Popoviciu [2] which is later proved partially by Rahman [1] and fully by Sudbery [3] states the following. Theorem 0.1. [1–3] (Popoviciu Conjecture/Rahman-Sudbery Theorem) Let n≥2and p(z) = (z−a1)(z−a2)· · · (z−an)∈C[z]be such that aj6=akfor some 1≤j, k ≤n, j 6=k. Then the polynomial q(z):=p(z)p0(z)· · · p(n−1)(z)∈C[z] has atleast n+ 1 distinct zeros. In this note, we formulate C*-algebraic analogue of Theorem 0.1 and verify it for degree 2 polynomials over commutative unital C*-algebras. Conjecture 0.2. (C*-algebraic Popoviciu Conjecture) Let Abe a unital commutative C*-algebra. Let n≥2and p(z)=(z−a1)(z−a2)· · · (z−an)∈ A[z]be such that aj6=akfor some 1≤j, k ≤n, j 6=k. Then the polynomial q(z):=p(z)p0(z)· · · p(n−1)(z)∈ A[z] has atleast n+ 1 distinct zeros. Theorem 0.3. Conjecture 0.2 holds for degree 2 polynomials. Proof. Let Abe a unital commutative C*-algebra. Let p(z) = (z−a)(z−b)∈ A[z] with a6=b. Then q(z) = p(z)p0(z)=(z−a)(z−b)(2z−(a+b)). Hence p(a) = p(b) = pa+b 2= 0. 1 K. MAHESH KRISHNA Note that a6=b, a 6=a+b 2, b 6=a+b 2.  References [1] Q. I. Rahman. The distinct zeros of the product of a polynomial and its successive derivatives. Can. Math. Bull., 14:267–269, 1971. [2] Q. I. Rahman and G. Schmeisser. Analytic theory of polynomials, volume 26 of Lond. Math. Soc. Monogr., New Ser. Oxford: Oxford University Press, 2002. [3] A. Sudbery. The number of distinct roots of a polynomial and its derivatives. Bull. Lond. Math. Soc., 5:13–17, 1973. 2