Automorphic equivalence problem for free associative algebras of rank two

Let K be the free associative algebra of rank 2 over an algebraically closed constructive field of any characteristic. We present an algorithm which decides whether or not two elements in K are equivalent under an automorphism of K. A modification of our algorithm solves the problem whether or not an element in K is a semiinvariant of a nontrivial automorphism. In particular, it determines whether or not the element has a nontrivial stabilizer in Aut K. An algorithm for equivalence of polynomials under automorphisms of C[x,y] was presented by Wightwick. Another, much simpler algorithm for automorphic equivalence of two polynomials in K[x,y] for any algebraically closed constructive field K was given by Makar-Limanov, Shpilrain, and Yu. In our approach we combine an idea of the latter three authors with an idea from the unpubished thesis of Lane used to describe automorphisms which stabilize elements of K. This also allows us to give a simple proof of the corresponding result for K[x,y] obtained by Makar-Limanov, Shpilrain, and Yu

Lifting vector bundles to Witt vector bundles

Let $p$ be a prime, and let $S$ be a scheme of characteristic $p$. Let $n \geq 2$ be an integer. Denote by $\mathbf{W}_n(S)$ the scheme of Witt vectors of length $n$, built out of $S$. The main objective of this paper concerns the question of extending (=lifting) vector bundles on $S$ to vector bundles on $\mathbf{W}_n(S)$. After introducing the formalism of Witt-Frobenius Modules and Witt vector bundles, we study two significant particular cases, for which the answer is positive: that of line bundles, and that of the tautological vector bundle of a projective space. We give several applications of our point of view to classical questions in deformation theory---see the Introduction for details. In particular, we show that the tautological vector bundle of the Grassmannian $Gr_{\mathbb{F}_p}(m,n)$ does not extend to $\mathbf{W}_2(Gr_{\mathbb{F}_p}(m,n))$, if $2 \leq m \leq n-2$. In the Appendix, we give algebraic details on our (new) approach to Witt vectors, using polynomial laws and divided powers. It is, we believe, very convenient to tackle lifting questions.Comment: Enriched version, with an appendi

Generalised polynomials and integer powers

We show that there does not exist a generalised polynomial which vanishes precisely on the set of powers of two. In fact, if $k \geq 2$ is and integer and $g \colon \mathbb{N} \to \mathbb{R}$ is a generalised polynomial such that $g(k^n) = 0$ for all $n \geq 0$ then there exists infinitely many $m \in \mathbb{N}$, not divisible by $k$, such that $g(mk^n) = 0$ for some $n \geq 0$. As a consequence, we obtain a complete characterisation of sequences which are simultaneously automatic and generalised polynomial.Comment: 51 page

Polynomial Bounds for Oscillation of Solutions of Fuchsian Systems

We study the problem of placing effective upper bounds for the number of zeros of solutions of Fuchsian systems on the Riemann sphere. The principal result is an explicit (non-uniform) upper bound, polynomially growing on the frontier of the class of Fuchsian systems of given dimension n having m singular points. As a function of n,m, this bound turns out to be double exponential in the precise sense explained in the paper. As a corollary, we obtain a solution of the so called restricted infinitesimal Hilbert 16th problem, an explicit upper bound for the number of isolated zeros of Abelian integrals which is polynomially growing as the Hamiltonian tends to the degeneracy locus. This improves the exponential bounds recently established by A. Glutsyuk and Yu. Ilyashenko.Comment: Will appear in Annales de l'institut Fourier vol. 60 (2010

Holographic Algorithm with Matchgates Is Universal for Planar #\#CSP Over Boolean Domain

We prove a complexity classification theorem that classifies all counting constraint satisfaction problems ($\#$CSP) over Boolean variables into exactly three categories: (1) Polynomial-time tractable; (2) $\#$P-hard for general instances, but solvable in polynomial-time over planar graphs; and (3) $\#$P-hard over planar graphs. The classification applies to all sets of local, not necessarily symmetric, constraint functions on Boolean variables that take complex values. It is shown that Valiant's holographic algorithm with matchgates is a universal strategy for all problems in category (2).Comment: 94 page