arXiv subjects
Publications and source records attributed to Mathieu Da Silva.
For $F \in \mathbb{Z}[s,t]$ a binary quadratic form which is irreducible over $\mathbb{Q}$, and $L$ an abelian number field with class number $1$, we obtain the order of magnitude for the number of values $F(s,t)$ which are a norm from $L$. Our result relies on the fundamental lemma of sieve theory and on geometry of numbers.
We study the proportion of conics given by $(\mathcal{C}_{\mathbf{F}, \mathbf{y}}) : F_0(\mathbf{y})x_0^2 + F_1(\mathbf{y})x_1^2 = F_2( \mathbf{y})x_2^2 $ which have a rational point $\mathbf{x} = (x_0 :x_1:x_2) \in \mathbb{P}^2(\mathbb{Q})$, where $\mathbf{y} = (y_0 : \dots : y_n)\in \mathbb{P}^n(\mathbb{Q})$ and $F_0,F_1,F_2 \in \mathbb{Z}[X_0,\ldots, X_n]$ are homogeneous polynomials in many variables of the same degree $d$. We provide an asymptotic formula for the number of $\mathbf{y}$ of bounded height such that the corresponding conic $(\mathcal{C}_{\mathbf{F}, \mathbf{y}})$ has a rational point. In particular, our result agrees with the Loughran--Smeets and the Loughran--Rome--Sofos conjectures. Our strategy is based on a recent result of Destagnol--Lyczak--Sofos relying on the circle method to estimate the average of an arithmetic function over polynomials in many variables. To this end, we study the proportion of conics $t_0x_0^2 + t_1x_1^2 + t_2x_2^2 = 0$ having a rational point, and coefficients $t_0,t_1,t_2$ in arithmetic progressions.
Energy consumption of IoT devices is a very important issue. For this reason, many techniques have been developed to allow IoT nodes to be aware of the amount of available energy. When energy is missing, the device halts and saves its state. One of those techniques is context saving, relying on the use of Non-Volatile Memories (NVM) to store and restore the state of the device. However, this information, as far as IoT devices deal with security, might be the target of attacks, including tampering and theft of confidential data. In this paper, we propose a SECure Context Saving (SECCS) approach that provides a context saving procedure and a hardware module easy to implement inside a System on Chip (SoC). This approach provides both confidentiality and integrity to all the CPU content saved into the target NVM.