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 andâ€¦ (More)

It is shown that the Jacobian Conjecture holds for all polynomial maps F : k â†’ k of the form F = x + H , such that JH is nilpotent and symmetric, when n â‰¤ 4. If H is also homogeneous a similar resultâ€¦ (More)

We show that the Minesweeper game is PP-hard, when the object is to locate all mines with the highest probability. When the probability of locating all mines may be infinitesimal, the Minesweeperâ€¦ (More)

In this paper, we show that the Jacobian conjecture holds for gradient maps in dimension n â‰¤ 3 over a field K of characteristic zero. We do this by extending the following result for n â‰¤ 2 by F.â€¦ (More)

It was conjectured by ÄŒernÃ½ in 1964, that a synchronizing DFA on n states always has a shortest synchronizing word of length at most (nâˆ’ 1), and he gave a sequence of DFAs for which this bound isâ€¦ (More)

The well-known ABC-conjecture is generally formulated as follows: The ABC-conjecture. Consider the set S of triples (A, B, C) âˆˆ N 3 such that ABC = 0, gcd{A, B, C} = 1 and A + B = C Then for every Ç«â€¦ (More)

It is known that strongly nilpotent matrices over a division ring are linearly triangularizable. We describe the structure of such matrices in terms of the strong nilpotency index. We apply ourâ€¦ (More)

Let k be a field of characteristic zero and F : k â†’ k a polynomial map of the form F = x + H, where H is homogeneous of degree d â‰¥ 2. We show that the Jacobian Conjecture is true for such mappings.â€¦ (More)

It was conjectured by \v{C}ern\'y 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 isâ€¦ (More)

In this paper we completely classify all polynomial maps of the form H = (u(x, y), v(x, y, z), h(u(x, y), v(x, y, z)) with JH nilpotent. AMS subject classification : 14R15, 14R10, 14E07.