##### Department of Mathematics,

University of California San Diego

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### Math 243 - Functional Analysis Seminar

## Jurij Volcic

#### Ben-Gurion University

## A Nullstellensatz for noncommutative polynomials: advances in determinantal representations

##### Abstract:

The talk concerns (matrices of) noncommutative polynomials $f=f(x)$ from the perspective of free real algebraic geometry. There are several natural notions of a ``zero set'' of $f$. The one we study is the \textbf{free locus} of $f$, $(f)$, which is defined to be the union of hypersurfaces $$\left\{X\in \operatorname{M}_n(k)^g:\det f(X)=0\right\}$$ over all $n\in\mathbb{N}$. The talk will describe a recent advance on characterizing when $(f_1) \subseteq (f_2)$ holds, which was mainly achieved using linear matrix pencils. The latter have been for decades an important tool in noncommutative algebra and other areas, e.g. linear systems, automata theory and computational complexity. Given a monic matrix pencil $L=I+\sum_jA_jx_j$, we can evaluate it at an arbitrary tuple of matrices $X$ as $$L(X)=I\otimes I+\sum_jA_j\otimes X_j.$$ The talk will be mostly concerned with singularity of these evaluations. First we will give an algebraic certificate for $(L_1)\subseteq (L_2)$ to hold. Then we will consider a fundamental irreducibility theorem for $(L)$ which is obtained with the aid of invariant theory. Next we will apply the preceding results to factorization in the free algebra. Lastly, smooth points on $(L)$ will be related to one-dimensional kernels of $L(X)$, which leads to the free version of Kippenhahn's conjecture and improves existing Positivstellens\''{a}tze on free semialgebraic sets.

Host: Bill Helton

### January 18, 2018

### 12:00 PM

### AP&M 6218

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