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Michiel de Bondt

Publications and source records attributed to Michiel de Bondt.

At least 37 records · Page 2Linked to original sources
arXiv2017

Quadratic polynomial maps with Jacobian rank two

Let $K$ be any field and $x = (x_1,x_2,\ldots,x_n)$. We classify all matrices $M \in {\rm Mat}_{m,n}(K[x])$ whose entries are polynomials of degree at most 1, for which ${\rm rk} M \le 2$. As a special case, we describe all such matrices $M$, which are the Jacobian matrix $J H$ (the matrix of partial derivatives) of a polynomial map $H$ from $K^n$ to $K^m$. Among other things, we show that up to composition with linear maps over $K$, $M = J H$ has only two nonzero columns or only three nonzero rows in this case. In addition, we show that ${\rm trdeg}_K K(H) = {\rm rk} J H$ for quadratic polynomial maps $H$ over $K$ such that $\frac12 \in K$ and ${\rm rk} J H \le 2$. Furthermore, we prove that up to conjugation with linear maps over $K$, nilpotent Jacobian matrices $N$ of quadratic polynomial maps, for which ${\rm rk} N \le 2$, are triangular (with zeroes on the diagonal), regardless of the characteristic of $K$. This generalizes several results by others. In addition, we prove the same result for Jacobian matrices $N$ of quadratic polynomial maps, for which $N^2 = 0$. This generalizes a result by others, namely the case where $\frac12 \in K$ and $N(0) = 0$.

math.AC↗
arXiv2017

Computations of Keller maps over fields with $\tfrac16$

We classify Keller maps $x + H$ in dimension $n$ over fields with $\tfrac16$, for which $H$ is homogeneous, and (1) deg $H = 3$ and rk $JH \le 2$; (2) deg $H = 3$ and $n \le 4$; (3) deg $H = 4$ and $n \le 3$; (4) deg $H = 4 = n$ and $H_1, H_2, H_3, H_4$ are linearly dependent over $K$. In our proof of these classifications, we formulate (and prove) several results which are more general than needed for these classifications. One of these results is the classification of all homogeneous polynomial maps $H$ as in (1) over fields with $\tfrac16$.

math.AG↗
arXiv2017

Quadratic homogeneous polynomial maps $H$ and Keller maps $x+H$ with $3 \le {\rm rk} J H \le 4$

We compute by hand all quadratic homogeneous polynomial maps $H$ and all Keller maps of the form $x + H$, for which ${\rm rk} J H = 3$, over a field of arbitrary characteristic. Furthermore, we use computer support to compute Keller maps of the form $x + H$ with ${\rm rk} J H = 4$, namely: $\bullet$ all such maps in dimension $5$ over fields with $\frac12$; $\bullet$ all such maps in dimension $6$ over fields without $\frac12$. We use these results to prove the following over fields of arbitrary characteristic: for Keller maps $x + H$ for which ${\rm rk} J H \le 4$, the rows of $J H$ are dependent over the base field.

math.AC↗
arXiv2017

Homogeneous quasi-translations in dimension 5

We give a proof in modern language of the following result by Paul Gordan and Max Nöther: 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 Nöther, 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 Nöther in $\mathbb P^4$' as well, but some of them assume that $h$ is irreducible, which Gordan and Nöther 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.

math.AG↗
arXiv2017

DFAs and PFAs with Long Shortest Synchronizing Word Length

It was conjectured by Černý in 1964, that a synchronizing DFA on $n$ states always has a shortest 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 $n \leq 4$, and with bounds on the number of symbols for $n \leq 10$. Here we give the full analysis for $n \leq 6$, without bounds on the number of symbols. For PFAs the bound is much higher. For $n \leq 6$ 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 give a construction of 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.

math.CO↗
arXiv2016

Irreducibility properties of Keller maps

Jedrzejewicz showed that a polynomial map over a field of characteristic zero is invertible, if and only if the corresponding endomorphism maps irreducible polynomials to irreducible polynomials. Furthermore, he showed that a polynomial map over a field of characteristic zero is a Keller map, if and only if the corresponding endomorphism maps irreducible polynomials to square-free polynomials. We show that the latter endomorphism maps other square-free polynomials to square-free polynomials as well. In connection with the above classification of invertible polynomial maps and the Jacobian Conjecture, we study irreducible properties of several types of Keller maps, to each of which the Jacobian Conjecture can be reduced. Herewith, we generalize the result of Bakalarski, that the components of cubic homogeneous Keller maps with a symmetric Jacobian matrix (over C and hence any field of characteristic zero) are irreducible. Furthermore, we show that the Jacobian Conjecture can even be reduced to any of these types with the extra condition that each affinely linear combination of the components of the polynomial map is irreducible. This is somewhat similar to reducing the planar Jacobian Conjecture to the so-called (planar) weak Jacobian Conjecture by Kaliman.

math.AG↗
arXiv2016

Mathieu subspaces of codimension less than n of Mat_n(K)

We classify all Mathieu subspaces of ${\rm Mat}_n(K)$ of codimension less than $n$, under the assumption that ${\rm char\,} K = 0$ or ${\rm char\,} K \ge n$. More precisely, we show that any proper Mathieu subspace of ${\rm Mat}_n(K)$ of codimension less than $n$ is a subspace of $\{M \in {\rm Mat}_n(K) \mid {\rm tr\,} M = 0\}$ if ${\rm char\,} K = 0$ or ${\rm char\,} K \ge n$. On the other hand, we show that every subspace of $\{M \in {\rm Mat}_n(K) \mid {\rm tr\,} M = 0\}$ of codimension less than $n$ in ${\rm Mat}_n(K)$ is a Mathieu subspace of ${\rm Mat}_n(K)$ if ${\rm char\,} K = 0$ or ${\rm char\,} K \ge n+1$.

math.RA↗
arXiv2015

Quasi-translations and singular Hessians

In 1876 in [8], the authors Paul Gordan and Max Nöther classify all homogeneous polynomials h in at most five variables for which the Hessian determinant vanishes. For that purpose, they study quasi-translations which are associated with singular Hessians. We will explain what quasi-translations are and formulate some elementary properties of them. Additionally, we classify all quasi-translations with Jacobian rank one and all so-called irreducible homogeneous quasi-translations with Jacobian rank two. The latter is an important result of [8]. Using these results, we classify all quasi-translations in dimension at most three and all homogeneous quasi-translations in dimension at most four. Furthermore, we describe the connection of quasi-translation with singular Hessians, and as an application, we will classify all polynomials in dimension two and all homogeneous polynomials in dimensions three and four whose Hessian determinant vanishes. More precisely, we will show that up to linear terms, these polynomials can be expressed in n-1 linear forms, where n is the dimension, according to an invalid theorem of Hesse. In the last section, we formulate some known results and conjectures in connection with quasi-translations and singular Hessians.

math.AG↗
arXiv2015

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 <= 3 variables such that det Hf <> 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.

math.AG↗
arXiv2014

Triangularization properties of power linear maps and the Structural Conjecture

In this paper, we discuss several additional properties a power linear Keller map may have. The Structural Conjecture by Druzkowski in [Dru] asserts that two such properties are equivalent, but we show that one of this properties is stronger than the other. We even show that the property of linear triangularizability is strictly in between. Furthermore, we give some positive results for small dimensions and small Jacobian ranks.

math.AG↗
arXiv2014

Some Remarks on the Jacobian Conjecture and Dru{ż}kowski mappings

In this paper, we first show that the Jacobian Conjecture is true for non-homogeneous power linear mappings under some conditions. Secondly, we prove an equivalent statement about the Jacobian Conjecture in dimension $r\geq 1$ and give some partial results for $r=2$. Finally, for a homogeneous power linear Keller map $F=X+H$ of degree $d \ge 2$, we give the inverse polynomial map under the condition that $JH^3=0$. We shall show that ${\operatorname{deg}}(F^{-1})\leq d^k$ if $k \le 2$ and $JH^{k+1}=0$, but also give an example with $d = 2$ and $JH^4=0$ such that ${\operatorname{deg}}(F^{-1})> d^3$.

math.AG↗
arXiv2013

Symmetric Jacobians

This article is about polynomial maps with a certain symmetry and/or antisymmetry in their Jacobians, and whether the Jacobian Conjecture is satisfied for such maps, or whether it is sufficient to prove the Jacobian Conjecture for such maps. For instance, we show that it suffices to prove the Jacobian conjecture for polynomial maps x + H over C such that JH satisfies all symmetries of the square, where H is homogeneous of arbitrary degree d >= 3.

math.AG↗
arXiv2013

The strong nilpotency index of a matrix

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 our results on quasi-translation x + H such that JH has strong nilpotency index two.

math.AG↗
arXiv2013

A few remarks on the Generalized Vanishing Conjecture

We show that the Generalized Vanishing Conjecture $$\forall_{m \ge 1} [\Lam^m f^m = 0] \Longrightarrow \forall_{m \gg 0} [\Lam^m (g f^m) = 0]$$ for a fixed differential operator $\Lam \in k[\partial]$ follows from a special case of it, namely that the additional factor $g$ is a power of the radical polynomial $f$. Next we show that in order to prove the Generalized Vanishing Conjecture (up to some bound on the degree of $\Lam$), we may assume that $\Lam$ is a linear combination of powers of distinct partial derivatives. At last, we show that the Generalized Vanishing Conjecture holds for products of linear forms in $\partial$, in particular homogeneous differential operators $Λ\in k[\partial_1,\partial_2]$.

math.AC↗
arXiv2012

The computational complexity of Minesweeper

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 Minesweeper game is even PSPACE-complete. In our construction, the player can reveal a boolean circuit in polynomial time, after guessing an initial square with no surrounding mines, a guess that has 99 percent probability of success. Subsequently, the mines must be located with a maximum probability of success. Furthermore, we show that determining the solvability of a partially uncovered Minesweeper board is NP-complete with hexagonal and triangular grids as well as a square grid, extending a similar result for square grids only by R. Kaye. Actually finding the mines with a maximum probability of success is again PP-hard or PSPACE-complete respectively. Our constructions are in such a way that the number of mines can be computed in polynomial time and hence a possible mine counter does not provide additional information. The results are obtained by replacing the dyadic gates in [3] by two primitives which makes life more easy in this context.

cs.CC↗
arXiv2012

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.

cs.CC↗
arXiv2012

Some remarks on the Jacobian conjecture and polynomial endomorphisms

In this paper, we first show that homogeneous Keller maps are injective on lines through the origin. We subsequently formulate a generalization, which is that under some conditions, a polynomial endomorphism with $r$ homogeneous parts of positive degree does not have $r$ times the same image point on a line through the origin, in case its Jacobian determinant does not vanish anywhere on that line. As a consequence, a Keller map of degree $r$ does not take the same values on $r > 1$ collinear points, provided $r$ is a unit in the base field. Next, we show that for invertible maps $x + H$ of degree $d$, such that $\ker \jac H$ has $n-r$ independent vectors over the base field, in particular for invertible power linear maps $x + (Ax)^{*d}$ with $\rk A = r$, the degree of the inverse of $x + H$ is at most $d^r$.

math.AG↗