Factorization of Noncommutative Polynomials and Nullstellensätze for the Free Algebra

This article gives a class of Nullstellensätze for noncommutative polynomials. The singularity set of a noncommutative polynomial f = f (x1, . . . , xg) is Z ( f ) = (Zn( f ))n, where Zn( f ) = {X ϵ Mn(C)g: detf (X) = 0}. The 1st main theorem of this article shows that the irreducible factors of f are in a natural bijective correspondence with irreducible components of Zn( 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: deth(X,X*) = 0}. A polynomial f that 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.