The works presented in this habilitation concern the algorithmics of polynomials. This is a central topic in computer algebra, with numerous applications both within and outside the field - cryptography, error-correcting codes, etc. For many problems, extremely efficient algorithms have been developed since the 1960s. Here, we are interested in how this efficiency is affected when space constraints are introduced. The first part focuses on the time-space complexity of fundamental polynomial computations - multiplication, division, interpolation, ... While naive algorithms typically have constant space complexity, fast algorithms generally require linear space. We develop algorithms that are both time- and space-efficient. This leads us to discuss and refine definitions of space complexity for function computation. In the second part, the space constraints are put on the inputs and outputs. Algorithms for polynomials assume in general a dense representation for the polynomials, that is storing the full list of coefficients. In contrast, we work with sparse polynomials, in which most coefficients vanish. In particular, we describe the first quasi-linear algorithm for sparse interpolation, which plays a role analogous to the Fast Fourier Transform in the sparse settings. We also explore computationally hard problems concerning divisibility and factorization of sparse polynomials.

翻译：本资格论文中呈现的研究工作涉及多项式算法学。这是计算机代数领域的核心课题，在密码学、纠错编码等学科内外均有广泛应用。自20世纪60年代以来，针对许多问题已发展出极其高效的算法。本文主要探讨引入空间约束时对算法效率的影响。第一部分聚焦于多项式基础运算（乘法、除法、插值等）的时间-空间复杂度：朴素算法通常具有常数空间复杂度，而快速算法往往需要线性空间。我们开发了兼具时间与空间效率的算法，并借此讨论和优化了函数计算中空间复杂度的定义。第二部分将空间约束置于输入与输出端：传统多项式算法通常采用稠密表示法（即存储完整的系数列表），而本研究针对稀疏多项式（即大多数系数为零的情况）展开。特别地，我们描述了稀疏插值领域的首个拟线性算法，该算法在稀疏场景中发挥着类似快速傅里叶变换的作用。同时，我们还探索了关于稀疏多项式可除性与因式分解的计算难题。