It was conjectured byČerný in 1964, that a synchronizing DFA on n states always has a synchronizing word of length at most (n−1) 2 , and he gave a sequence of DFAs for which this bound is reached. Until now a full analysis of all DFAs reaching this bound was only given for n ≤ 4, and with bounds on the number of symbols for n ≤ 10. Here we give the full analysis for n ≤ 6, without bounds on the number of symbols. For PFAs on n ≤ 6 states we do a similar analysis as for DFAs and find the maximal shortest synchronizing word lengths, exceeding (n − 1) 2 for n = 4, 5, 6. For arbitrary n we use rewrite systems to construct a PFA on three symbols with exponential shortest synchronizing word length, giving significantly better bounds than earlier exponential constructions. We give a transformation of this PFA to a PFA on two symbols keeping exponential shortest synchronizing word length, yielding a better bound than applying a similar known transformation.

Let k be a field of characteristic zero. For small n, we classify all f ∈ k [n] such that the Hessian of f is singular.

Abstract. The main result of this paper asserts that it suffices to prove the Jacobian Conjecture for all polynomial maps of the form x + H, where H is homogeneous (of degree 3) and JH is nilpotent and symmetric. Also a 6-dimensional counterexample is given to a dependence problem posed by de Bondt and van den Essen (2003).

It was conjectured by Černý in 1964, that a synchronizing DFA on [Formula: see text] states always has a synchronizing word of length at most [Formula: see text], and he gave a sequence of DFAs for which this bound is reached. Until now a full analysis of all DFAs reaching this bound was only given for [Formula: see text], and with bounds on the number of symbols for [Formula: see text]. Here we give the full analysis for [Formula: see text], without bounds on the number of symbols. For PFAs (partial automata) on [Formula: see text] states we do a similar analysis as for DFAs and find the maximal shortest synchronizing word lengths, exceeding [Formula: see text] for [Formula: see text]. Where DFAs with long synchronization typically have very few symbols, for PFAs we observe that more symbols may increase the synchronizing word length. For PFAs on [Formula: see text] states and two symbols we investigate all occurring synchronizing word lengths. We give series of PFAs on two and three symbols, reaching the maximal possible length for some small values of [Formula: see text]. For [Formula: see text], the construction on two symbols is the unique one reaching the maximal length. For both series the growth is faster than [Formula: see text], although still quadratic. Based on string rewriting, for arbitrary size we construct a PFA on three symbols with exponential shortest synchronizing word length, giving significantly better bounds than earlier exponential constructions. We give a transformation of this PFA to a PFA on two symbols keeping exponential shortest synchronizing word length, yielding a better bound than applying a similar known transformation. Both PFAs are transitive. Finally, we show that exponential lengths are even possible with just one single undefined transition, again with transitive constructions.

Let k be a field of characteristic zero and F : k 3 → k 3 a polynomial map of the form F = x + H , where H is homogeneous of degree d 2. We show that the Jacobian Conjecture is true for such mappings. More precisely, we show that if J H is nilpotent there exists an invertible linear map T such that T −1 H T = (0, h 2 (x 1 ), h 3 (x 1 , x 2 )), where the h i are homogeneous of degree d. As a consequence of this result, we show that all generalized Drużkowski mappings F = x + H = (x 1 + L d 1 , . . . , x n + L d n ), where L i are linear forms for all i and d 2, are linearly triangularizable if J H is nilpotent and rk J H 3.