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Lower bounds and reconstruction algorithms for sums of affine powers

Abstract : The general framework of this thesis is the study of polynomials as objects of models of computation. This approach allows to define precisely the evaluation complexity of a polynomial, and then to classify families of polynomials depending on their complexity. In this thesis, we focus on the study of the model of sums of affine powers, that is polynomials that can be written as f = ∑_{i = 1}^s α_i ℓ_i^{e_i}, with deg ℓ_i=1. This model is quite natural, as it extends both the Waring model f = ∑ α_i ℓ_i^d, and the sparsest shift model f = ∑ α_i ℓ^{e_i}, but it is still not well known. In this work, we obtained structural results for the univariate variant of this model, which allow us to obtain lower bounds and reconstruction algorithms, that solve the following problem : given f = ∑ α_i (x-a_i)^{e_i} as a list of its coefficient, find the values of the α_i’s, e_i’s and a_i’s in the optimal decomposition of f. We also studied the multivariate case and obtained several reconstruction algorithms that work whenever the number of terms in the optimal expression is small in terms of the number of variable or the degree of the polynomial.
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Submitted on : Tuesday, October 16, 2018 - 10:39:07 AM
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Timothée Pecatte. Lower bounds and reconstruction algorithms for sums of affine powers. Computational Complexity [cs.CC]. Université de Lyon, 2018. English. ⟨NNT : 2018LYSEN029⟩. ⟨tel-01896437⟩

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