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Gerald Bourgeois

Publications and source records attributed to Gerald Bourgeois.

12 recordsLinked to original sources
arXiv2015

Nonsymmetric generic matrix equations

Let $(A_i)_{0\leq i\leq k}$ be generic matrices over $\mathbb{Q}$, the field of rational numbers. Let $K=\mathbb{Q}(E)$, where $E$ denotes the entries of the $(A_i)_i$, and let $\overline{K}$ be the algebraic closure of $K$. We show that the generic unilateral equation $A_kX^k+\cdots+A_1X+A_0=0_n$ has $\binom{nk}{n}$ solutions $X\in\mathcal{M}_n(\overline{K})$. Solving the previous equation is equivalent to solving a polynomial of degree $kn$, with Galois group $S_{kn}$ over $K$. Let $(B_i)_{i\leq k}$ be fixed $n\times n$ matrices with entries in a field $L$. We show that, for a generic $C\in\mathcal{M}_n(L)$, a polynomial equation $g(B_1,\cdots,B_k,X)=C$ admits a finite fixed number of solutions and these solutions are simple. We study, when $n=2$, the generic non-unilateral equations $X^2+BXC+D=0_2$ and $X^2+BXB+C=0_2$. We consider the unilateral equation $X^k+C_{k-1}X^{k-1}+\cdots+C_1X+C_0=0_n$ when the $(C_i)_i$ are $n\times n$ generic commuting matrices ; we show that every solution $X\in\mathcal{M}_n(\overline{K})$ commutes with the $(C_i)_i$. When $n=2$, we prove that the generic equation $A_1XA_2X+XA_3X+X^2A_4+A_5X+A_6=0_2$ admits $16$ isolated solutions in $\mathcal{M}_2(\overline{K})$, that is, according to the Bézout's theorem, the maximum for a quadratic $2\times 2$ matrix equation.

math.RA↗
arXiv2014

The matrix equations $XA-AX=X^αg(X)$ over fields or rings

Let $n,α\geq 2$. Let $K$ be an algebraically closed field with characteristic $0$ or greater than $n$. We show that the dimension of the variety of pairs $(A,B)\in {M_n(K)}^2$, with $B$ nilpotent, that satisfy $AB-BA=A^α$ or $A^2-2AB+B^2=0$ is $n^2-1$ ; moreover such matrices $(A,B)$ are simultaneously triangularizable. Let $R$ be a reduced ring such that $n!$ is not a zero-divisor and $A$ be a generic matrix over $R$ ; we show that $X=0$ is the sole solution of $AX-XA=X^α$. Let $R$ be a commutative ring with unity ; let $A$ be similar to $\mathrm{diag}(λ_1I_{n_1},\cdots,λ_rI_{n_r})$ such that, for every $i\not= j$, $λ_i-λ_j$ is not a zero-divisor. If $X$ is a nilpotent solution of $XA-AX=X^αg(X)$ where $g$ is a polynomial, then $AX=XA$.

math.RA↗
arXiv2014

The matrix equation $XA-AX=f(X)$ when $A$ is diagonalizable

$K$ is an algebraically closed field with characteristic $0$ and $f$ is a polynomial or a holomorphic function. We study all solutions of the equation $XA-AX=f(X)$, in the unknown $X\in M_n(K)$, when $A\in M_n(K)$ is diagonalizable.

math.RA↗
arXiv2012

Common Invariant Subspace and Commuting Matrices

Let $K$ be a perfect field, $L$ be an extension field of $K$ and $A,B\in\mathcal{M}_n(K)$. If $A$ has $n$ distinct eigenvalues in $L$ that are explicitly known, then we can check if $A,B$ are simultaneously triangularizable over $L$. Now we assume that $A,B$ have a common invariant proper vector subspace of dimension $k$ over an extension field of $K$ and that $χ_A$, the characteristic polynomial of $A$, is irreducible over $K$. Let $G$ be the Galois group of $χ_A$. We show the following results i) If $k\in{1,n-1}$, then $A,B$ commute. ii) If $1\leq k\leq n-1$ and $G=\mathcal{S}_n$ or $G=\mathcal{A}_n$, then $AB=BA$. iii) If $1\leq k\leq n-1$ and $n$ is a prime number, then $AB=BA$. Yet, when $n=4,k=2$, we show that $A,B$ do not necessarily commute if $G$ is not $\mathcal{S}_4$ or $\mathcal{A}_4$. Finally we apply the previous results to solving a matrix equation.

math.RA↗
arXiv2012

About the matrix function X->AX+XA

Let K be an infinite field such that its characteristic is not 2. We show that, for every $A\in\mathcal{M}_n(K)$ such that $\mathrm{rank}(A)\geq n/2$, there exists $B\in\mathcal{M}_n(K)$ such that $B$ is similar to $A$ and $A+B$ is invertible. Let $K$ be a subfield of $\mathbb{R}$. We show that, if $n$ is even, then for every $X\in\mathcal{M}_n(K)$, $\det(AX+XA)\geq 0$ if and only if either $\mathrm{rank}(A)<n/2$ or there exists $α\in K,α\leq 0$, such that $A^2=αI_n$.

math.RA↗
arXiv2012

How to solve the matrix equation XA-AX=f(X)

Let f be an analytic function defined on a complex domain Omega and A be a (n,n) complex matrix. We assume that there exists a unique alpha satisfying f(alpha)=0. When f'(alpha)=0 and A is non derogatory, we solve completely the equation XA-AX=f(X). This generalizes Burde's results. When f'(alpha) is not zero, we give a method to solve completely the equation XA-AX=f(X): we reduce the problem to solve a sequence of Sylvester equations. Solutions of the equation f(XA-AX)=X are also given in particular cases.

math.RA↗
arXiv2012

Similar Powers of a Matrix

Let $p,q$ be coprime integers such that $|p|+|q|>2$. We characterize the matrices $A\in\mathcal{M}_n(\mathbb{C})$ such that $A^p$ and $A^q$ are similar. If $A$ is invertible, we prove that $A$ is a polynomial in $A^p$ and $A^q$. To achieve this, we study the matrix equation $B^{-1}A^pB=A^q$. We show that for such matrices, $B^{-1}AB$ and $A$ commute. When $A$ is diagonalizable, $A$ is a root of $I_n$ and $B^{-1}AB$ is a power of $A$. We explicitly solve the previous equation when $A$ has $n$ distinct eigenvalues or when $A$ has a sole eigenvalue. In the second part, we completely solve the $2\times{2}$ case of the more general matrix equation $A^{r}B^{s}A^{r'}B^{s'}=\pm{I}_2$.

math.RA↗
arXiv2011

Commmuting exponentials in dimension at most 3

Let A,B be two square complex matrices of dimension at most 3. We show that the following conditions are equivalent i) There exists a finite subset U included in {2,3,4,...} such that for every positive integer t that is not in U, exp(tA+B)=exp(tA)exp(B)=exp(B)exp(tA). ii) The pair (A,B) has property L of Motzkin and Taussky and exp(A+B)=exp(A)exp(B)=exp(B)exp(A).

math.RA↗
arXiv2011

What about A,B if AB-BA and A commute

Let A,B be complex n,n complex matrices such that AB-BA and A commute. We show that, if n=2 then A,B are simultaneously triangularizable and if n>=3 then there exists such a couple A,B such that the pair (A,B) has not property L of Motzkin-Taussky and such that B and C are not simultaneously triangularizable.

math.RA↗
arXiv2009

Dynamical Systems of Simplices in Dimension 2 or 3

Let T_0=(A_0,..,A_d) be a d-simplex, G_0 its centroid, S its circumsphere, O the center of S. Let (B_i) be the points where S intersects the lines (G_0A_i), T_1 the d-simplex (B_1,..,B_d), and G_1 its centroid. By iterating this construction, a dynamical system of d-simplices (T_i) with centroids (G_i) is constructed. For d=2 or 3, we prove that the sequence (OG_i) is decreasing and tends to 0. We consider the sequences (T_{2i})_i and (T_{2i+1})_i ; for d=2 they converge to two equilateral triangles with at least quadratic speed ; for d=3 they converge to two isosceles tetrahedra with at least geometric speed. In this last case, we give an explicit expression of the lengths of the edges of the limit form. We show also that if T_0 is a planar cyclic quadrilateral then (T_n) converges to a rectangle with at least geometric speed or eventually to a square with a speed that is conjectured as cubic. The proofs are largely algebraic and use Grobner basis computations.

math.DS↗
arXiv2006

Attaque algebrique de NTRU a l'aide des vecteurs de Witt

One improves an algebraic attack of NTRU due to Silverman, Smart and Vercauteren; the latter considered the first 2 bits of a Witt vector attached to the research of the secret key; here the first 4 bits are considered, which provides additional equations of degrees 4 and 8.

cs.CR↗