Factorization of noncommutative polynomials and Nullstellensätze for the free algebra

Abstract

This article gives a class of Nullstellensätze for noncommutative polynomials. The singularity set of a noncommutative polynomial f = f (x 1 , . . . , x g ) isThe first main theorem of this article shows that the irreducible factors of f are in a natural bijective correspondence with irreducible components of Z n (f ) for every sufficiently large n.With each polynomial h in x and x * one also associates its real singularity set Z re (h) = {X : det h(X, X * ) = 0}. A polynomial f which depends on x alone (no x * variables) will be called analytic. The main Nullstellensatz proved here is as follows: for analytic f but for h dependent on possibly both x and x * , the containment Z (f ) ⊆ Z re (h) is equivalent to each factor of f being "stably associated" to a factor of h or of h * .For perspective, classical Hilbert type Nullstellensätze typically apply only to analytic polynomials f, h, while real Nullstellensätze typically require adjusting the functions by sums of squares of polynomials (sos). Since the above "algebraic certificate" does not involve a sos, it seems justified to think of this as the natural determinantal Hilbert Nullstellensatz. An earlier paper of the authors (Adv. Math. 331 (2018) 589-626) obtained such a theorem for special classes of analytic polynomials f and h. This paper requires few hypotheses and hopefully brings this type of Nullstellensatz to near final form.Finally, the paper gives a Nullstellensatz for zeros V(f ) = {X : f (X, X * ) = 0} of a hermitian polynomial f , leading to a strong Positivstellensatz for quadratic free semialgebraic sets by the use of a slack variable.