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 $f=f(x_1,\dots ,x_g)$ is $\mathscr{Z}(\,f)=(\mathscr{Z}_n(\,f))_n$, where $\mathscr{Z}_n(\,f)=\{X \in{\operatorname{M}}_{n}({\mathbb{C}})^g \colon \det f(X) = 0\}.$ The 1st main theorem of this article shows that the irreducible factors of $f$ are in a natural bijective correspondence with irreducible components of $\mathscr{Z}_n(\,f)$ for every sufficiently large $n$. With each polynomial $h$ in $x$ and $x^*$ one also associates its real singularity set $\mathscr{Z}^{{\operatorname{re}}}(h)=\{X\colon \det h(X,X^*)=0\}$. 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 $\mathscr{Z}(\,f) \subseteq \mathscr{Z}^{{\operatorname{re}}} (h)$ 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 ${\mathcal{V}}(\,f)=\{X\colon f(X,X^*)=0\}$ of a hermitian polynomial $f$, leading to a strong Positivstellensatz for quadratic free semialgebraic sets by the use of a slack variable.

‘Unforced’ Navier–Stokes solutions derived from convection in a curved channel

Steady Boussinesq flow in a weakly curved channel driven by a horizontal temperature gradient is considered. Linear variation in the transverse direction is assumed so that the problem reduces to a system of ordinary differential equations. A series expansion in $G$, a parameter proportional to the Grashof number and the square root of the curvature, reveals a real singularity and anticipates hysteresis. Numerical solutions are found using path continuation and the bifurcation diagrams for different parameter values are obtained. Multivalued solutions are observed as $G$ and the Prandtl number vary. Often fields with the imposed structure that satisfy all the governing equations are insensitive to the boundary conditions and can be regarded as perturbations of the homogeneous (or ‘unforced’) problem. Four such unforced solutions are found. In two of these the velocity remains coupled with temperature which, formally, scales as $1/G$ as $G\rightarrow 0$. The other two are purely hydrodynamic. The existence of such solutions is due to the unbounded nature of the domain. It is shown that these occur not only for the Dean equations, but constitute previously unreported solutions of the full Navier–Stokes equations in an annulus of arbitrary curvature. Two additional unforced solutions are found for large curvature.

Analysis of Stress on Interface Crack of Orthotropic and Isotropic Bi-Materials

This paper is concerned in semi-infinite interface crack of orthotropic and isotropic bi-materials and using the composite material fracture complex function method. By means of constructing special stress functions with two real singularity index and solving the problem of a class of generalized bi-harmonic equations , the stress and displacement fields of two dissimilar materials are obtained .Results demonstrate that the stress and displacement fields near the crack tip show mixed crack characteristics without oscillation.

Caustics of the harmonic double cusp near an E 6 point

The harmonic double cusp is the real singularity x 4 – 6 x 2 y 2 + y 4 . Its unfolding is eight-dimensional and contains a stratum of codimension five corresponding to the E 6 singularity x 3 + y 4 . A complete description of the bifurcation set in a neighbourhood of the E 6 stratum is given by means of a tableau of two-dimensional sections over the three-dimensional base. The tableau also gives information about the bifurcation set near other strata of high codimension, notably D – 6 : x 2 y – y 5 and A * 7 : x 2 – y 8 , and it shows how these strata fit together. In optics, the harmonic double cusp organizes a large family of caustics produced by generic thin lenses. Some predictions made by the tableau have already been confirmed by observations of caustics from liquid drops: points in the tableau base correspond to experimental set-ups, and the fibre corresponds to the viewing screen.