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              <p>Let $\K$ be a field and $(f_1, \ldots, f_n)\subset \K[X_1, \ldots, X_n]$ be a sequence of quasi-homogeneous polynomials of respective weighted degrees $(d_1, \ldots, d_n)$ w.r.t a system of weights $(w_{1},\dots,w_{n})$. Such systems are likely to arise from a lot of applications, including physics or cryptography. We design strategies for computing Gröbner bases for quasi-homogeneous systems by adapting existing algorithms for homogeneous systems to the quasi-homogeneous case. Overall, under genericity assumptions, we show that for a generic zero-dimensional quasi-homogeneous system, the complexity of the full strategy is polynomial in the weighted Bézout bound $\prod_{i=1}^{n}d_{i} / \prod_{i=1}^{n}w_{i}$. We provide some experimental results based on generic systems as well as systems arising from a cryptography problem. They show that taking advantage of the quasi-homogeneous structure of the systems allow us to solve systems that were out of reach otherwise.</p>
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              <p>Soit $\K$ un corps et $(f_1,\ldots,f_n) \subset \K[X_1,\ldots,X_n]$ une famille de polynômes quasi-homogènes de degrés respectifs $(d_1,\dots,d_n)$ pour un système de poids $(w_1,\dots,w_n)$. De tels systèmes apparaissent dans de nombreuses applications, issus par exemple de la physique ou de la cryptographie. On donne une stratégie pour calculer la base de Gröbner de systèmes quasi-homogènes, en adaptant les algorithmes qui existent déjà pour les systèmes homogènes, au cas quasi-homogène. Sous des hypothèses de généricité, on montre que pour un système quasi-homogène de dimension nulle, la complexité de la stratégie est polynomiale en la borne de Bézout pondérée $\prod_{i=1}^{n}d_{i} / \prod_{i=1}^{n}w_{i}$. On présente des résultats expérimentaux obtenus pour des systèmes génériques, ainsi que pour des systèmes apparaissant dans un problème issu de la cryptographie. Ces expériences montrent que l'exploitation de la structure quasi-homogène des systèmes nous permet de résoudre des problèmes qui étaient auparavant hors de portée.</p>
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