Positive trace polynomials and the universal Procesi-Schacher conjecture

The Procesi–schacher Conjecture and Hilbert’s 17th Problem for Algebras with Involution

In 1976 Procesi and Schacher developed an Artin–Schreier type theory for central simple algebras with involution and conjectured that in such an algebra a totally positive element is always a sum of hermitian squares. In this paper elementary counterexamples to this conjecture are constructed and cases are studied where the conjecture does hold. Also, a Positivstellensatz is established for non...

Universal Bernoulli Polynomials And

Test polynomials, retracts, and the Jacobian conjecture

Let K[x, y] be the algebra of two-variable polynomials over a field K. A polynomial p = p(x, y) is called a test polynomial for automorphisms if, whenever φ(p) = p for a mapping φ of K[x, y], this φ must be an automorphism. Here we show that p ∈ C[x, y] is a test polynomial if and only if p does not belong to any proper retract of C[x, y]. This has the following corollary that may have applicat...

Hessian Nilpotent Polynomials and the Jacobian Conjecture

Let z = (z1, · · · , zn) and ∆ = ∑n i=1 ∂ 2 ∂z i the Laplace operator. The main goal of the paper is to show that the wellknown Jacobian conjecture without any additional conditions is equivalent to the following what we call vanishing conjecture: for any homogeneous polynomial P (z) of degree d = 4, if ∆P(z) = 0 for all m ≥ 1, then ∆P(z) = 0 when m >> 0, or equivalently, ∆P(z) = 0 when m > 3 2...

Trace-positive polynomials, sums of hermitian squares and the tracial moment problem

A polynomial f in non-commuting variables is trace-positive if the trace of f(A) is positive for all tuples A of symmetric matrices of the same size. The investigation of trace-positive polynomials and of the question of when they can be written as a sum of hermitian squares and commutators of polynomials are motivated by their connection to two famous conjectures: The BMV conjecture from stati...