Solving Mahjong Solitaire boards with peeking

We first prove that solving Mahjong Solitaire boards with peeking is NP-complete, even if one only allows isolated stacks of the forms /aab/ and /abb/. We subsequently show that layouts of isolated stacks of heights one and two can always be solved with peeking, and that doing so is in P, as well as finding an optimal algorithm for such layouts without peeking. Next, we describe a practical algorithm for solving Mahjong Solitaire boards with peeking, which is simple and fast. The algorithm uses an effective pruning criterion and a heuristic to find and prioritize critical groups. The ideas of the algorithm can also be applied to solving Shisen-Sho with peeking.Comment: 10 page

Polynomials with constant Hessian determinants in dimension three

In this paper, we show that the Jacobian conjecture holds for gradient maps in dimension n <= 3 over a field K of characteristic zero. We do this by extending the following result for n <= 2 by F. Dillen to n <= 3: if f is a polynomial of degree larger than two in n <= 3 variables such that the Hessian determinant of f is constant, then after a suitable linear transformation (replacing f by f(Tx) for some T in GL_n(K)), the Hessian matrix of f becomes zero below the anti-diagonal. The result does not hold for larger n. The proof of the case det Hf in K* is based on the following result, which in turn is based on the already known case det Hf = 0: if f is a polynomial in n 0, then after a suitable linear transformation, there exists a positive weight function w on the variables such that the Hessian determinant of the w-leading part of f is nonzero. This result does not hold for larger n either (even if we replace `positive' by `nontrivial' above). In the last section, we show that the Jacobian conjecture holds for gradient maps over the reals whose linear part is the identity map, by proving that such gradient maps are translations (i.e. have degree 1) if they satisfy the Keller condition. We do this by showing that this problem is the polynomial case of the main result of [Pog]. For polynomials in dimension n <= 3, we generalize this result to arbitrary fields of characteristic zero.Comment: 16 pages, refereed for JPAA, stripped w.r.t. previous version in favor of a wider audienc

Homogeneous quasi-translations in dimension 5

We give a proof in modern language of the following result by Paul Gordan and Max N\"other: a homogeneous quasi-translation in dimension $5$ without linear invariants would be linearly conjugate to another such quasi-translation $x + H$, for which $H_5$ is algebraically independent over $\mathbb C$ of $H_1, H_2, H_3, H_4$. Just like Gordan and N\"other, we apply this result to classify all homogeneous polynomials $h$ in $5$ indeterminates from which the Hessian determinant is zero. Others claim to have reproved 'the result of Gordan and N\"other in $\mathbb P^4$' as well, but some of them assume that $h$ is irreducible, which Gordan and N\"other did not. Furthermore, they do not use the above result about homogeneous quasi-translations in dimension $5$ for their classifications. (There is however one paper which could use this result very well, to fix a gap caused by an error.) We derive some other properties which $H$ would have. One of them is that ${\rm deg}\, H \ge 15$, for which we give a proof which is less computational than another proof of it by Dayan Liu. Furthermore, we show that the Zariski closure of the image of $H$ would be an irreducible component of $V(H)$, and prove that every other irreducible component of $V(H)$ would be a $3$-dimensional linear subspace of $\mathbb C^5$ which contains the fifth standard basis unit vector.Comment: 34 page

Some remarks on Mathieu subspaces over associative algebras

In this paper, we generalize some of the results of [9], and add some new results. Furthermore, we take a closer look at strongly simple algebras, which are introduced in [9].Comment: 27 pages; result numbers in section 2 and the end of section 3 have changed w.r.t v