Les amibesdes variétés algébriques dans ${\left({ℂ}^{*}\right)}^n$ sont les images de ces variétés par l’application des moments $\mathrm{Log}:{\left({ℂ}^{*}\right)}^n\to {ℝ}^n$, $\mathrm{Log}:(z_1,...,z_n)\mapsto (log|z_1|,...,log|z_n\left|\right)$. Des résultats obtenus par G. Mikhalkin montrent l’utilité des amibes pour l’étude des variétés algébriques réelles et complexes. Les amibes peuvent être déformées en des complexes polyédraux appelésvariétés algébriques tropicales. Cette déformation permet, en particulier, de calculer les invariants de Gromov-Witten du plan projectif et d’autres surfaces toriques en dénombrant des courbes...

A closed loop parametrical identification procedure for continuous-time constant linear systems is introduced. This approach which exhibits good robustness properties with respect to a large variety of additive perturbations is based on the following mathematical tools: (1) module theory; (2) differential algebra; (3) operational calculus. Several concrete case-studies with computer simulations demonstrate the efficiency of our on-line identification scheme.

A closed loop parametrical identification procedure for continuous-time constant linear systems is introduced. This approach which exhibits good robustness properties with respect to a large variety of additive perturbations is based on the following mathematical tools: (1) module theory; (2) differential algebra; (3) operational calculus. Several concrete case-studies with computer simulations demonstrate the efficiency of our on-line identification scheme.

Let $N/K$ be a Galois extension with Galois group $𝒢$. We study the set $𝒯\left(𝒢\right)$ of $\mathbb{Z}$-linear combinations of characters in the Burnside ring $ℬ\left(𝒢\right)$ which give rise to $\mathbb{Z}$-linear combinations of trace forms of subextensions of $N/K$ which are trivial in the Witt ring W$\left(K\right)$ of $K$. In particular, we prove that the torsion subgroup of $ℬ\left(𝒢\right)/𝒯\left(𝒢\right)$ coincides with the kernel of the total signature homomorphism.

Let L/K be a finite Galois extension of complete discrete valued fields of characteristic p. Assume that the induced residue field extension $k_L/k_K$ is separable. For an integer n ≥ 0, let $W_n{(}_L)$ denote the ring of Witt vectors of length n with coefficients in ${}_L$. We show that the proabelian group ${H^1(G,W_n{(}_L))}_{n\in \mathbb{N}}$ is zero. This is an equicharacteristic analogue of Hesselholt’s conjecture, which was proved before when the discrete valued fields are of mixed characteristic.

In this paper we give a term equivalence between the simple k-cyclic Post algebra of order p, L p,k, and the finite field F(p k) with constants F(p). By using Lagrange polynomials, we give an explicit procedure to obtain an interpretation Φ1 of the variety V(L p,k) generated by L p,k into the variety V(F(p k)) generated by F(p k) and an interpretation Φ2 of V(F(p k)) into V(L p,k) such that Φ2Φ1(B) = B for every B ε V(L p,k) and Φ1Φ2(R) = R for every R ε V(F(p k)).

Using the techniques developed by Jean Ecalle for the study of nonlinear differential equations, we prove that the $q$-difference equation$$x{\sigma }_{q}y=y+b(y,x)$$with $\left({\sigma }_{q}f\right)\left(x\right)=f\left(qx\right)$ ($q\>1$) and $b(0,0)={\partial }_{y}b(0,0)=0$ is analytically conjugated to one of the following equations :$$x{\sigma }_{q}y=y\text{ou}x{\sigma }_{q}y=y+x$$