• Corpus ID: 249889951

C*-algebraic Casas-Alvero Conjecture

  title={C*-algebraic Casas-Alvero Conjecture},
  author={K. Mahesh Krishna},
: Based on Casas-Alvero conjecture [J. Algebra, 2001] we formulate the following conjecture. C*-algebraic Casas-Alvero Conjecture : Let A be a commutative C*-algebra, n ∈ N and let P ( z ) := ( z − a 1 )( z − a 2 ) · · · ( z − a n ) be a polynomial over A with a 1 , a 2 , . . . , a n ∈ A . If P shares a common zero with each of its (first) n − 1 derivatives, then it is n th power of a linear monic C*-algebraic polynomial. We show that C*-algebraic Casas-Alvero Conjecture holds for C*-algebraic… 



C*-algebraic Schoenberg Conjecture

: Based on Schoenberg conjecture [Amer. Math. Monthly., 1986] /Malamud-Pereira theorem [J. Math. Anal. Appl, 2003] , [Trans. Amer. Math. Soc., 2005] we formulate the following conjecture which we

On some properties of the Abel–Goncharov polynomials and the Casas-Alvero problem

ABSTRACT We derive new properties of the Abel–Goncharov interpolation polynomials, relating them to investigate necessary and sufficient conditions for an arbitrary polynomial of degree n to be

Convex Hulls and The Casas-Alvero Conjecture for the Complex Plane

It has been conjectured by Casas-Alvero that polynomials of degree n over fields of characteristic 0, share roots with each of its n − 1 derivatives if and only if those polynomials have one root of

Polynomial problems of the Casas-Alvero type

We establish necessary and sufficient conditions for an arbitrary polynomial of degree $n$, especially with only real roots, to be trivial, i.e. to have the form a(x-b)^n. To do this, we derive new

On the Casas-Alvero conjecture

The Casas–Alvero conjecture says that a complex univariate polynomial having roots in common with each of its derivatives must be a power of a linear polynomial. In this expository note we review

C*-algebraic Gauss-Lucas Theorem and C*-algebraic Sendov's Conjecture

Using a result of Robertson [Proc. Edinburgh Math. Soc. (2), 1976], we introduce a notion of differentiation of maps on certain classes of unital commutative C*-algebras. We then derive C*-algebraic

C*-algebraic Smale Mean Value Conjecture and Dubinin-Sugawa Dual Mean Value Conjecture

: Based on Smale mean value conjecture [Bull. Amer. Math. Soc., 1981] and Dubinin-Sugawa dual mean value conjecture [Proc. Japan Acad. Ser. A Math. Sci., 2009] we formulate the following conjectures.

Constraints on counterexamples to the Casas-Alvero conjecture and a verification in degree 12

A number of constraints on hypothetical counterexamples to the Casas-Alvero conjecture are proved, building on ideas of Graf von Bothmer, Labs, Schicho and van de Woestijne that were recently reinterpreted by Draisma and de Jong in terms of p-adic valuations.

Around some extensions of Casas-Alvero conjecture for non-polynomial functions

We show that two natural extensions of the real Casas-Alvero conjecture in the nonpolynomial setting do not hold.