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Gr{\"o}bner bases is one the most powerful tools in algorithmic non-linearalgebra. Their computation is an intrinsically hard problem with a complexityat least single exponential in the number of variables. However, in most of thecases, the polynomial systems coming from applications have some kind ofstructure. For example , several problems in computer-aided design, robotics,vision, biology , kinematics, cryptography, and optimization involve sparsesystems where the input polynomials have a few non-zero terms. Our approach toexploit sparsity is to embed the systems in a semigroup algebra and to computeGr{\"o}bner bases over this algebra. Up to now, the algorithms that follow thisapproach benefit from the sparsity only in the case where all the polynomialshave the same sparsity structure, that is the same Newton polytope. Weintroduce the first algorithm that overcomes this restriction. Under regularityassumptions, it performs no redundant computations. Further, we extend thisalgorithm to compute Gr{\"o}bner basis in the standard algebra and solve sparsepolynomials systems over the torus $(C*)^n$. The complexity of the algorithmdepends on the Newton polytopes.

Gr{\"o}bner Basis over Semigroup Algebras: Algorithms and Applications for Sparse Polynomial Systems