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：本资格论文的研究工作涉及多项式的算法领域。这是计算机代数中的核心课题，在学科内外（如密码学、纠错码等）均有广泛应用。自20世纪60年代以来，针对许多问题已开发出极为高效的算法。本文关注的是引入空间约束后对效率产生的影响。第一部分聚焦于基本多项式运算（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，而快速算法则通常需要线性空间。我们开发了同时兼顾时间与空间效率的算法，并由此对函数计算空间复杂度的定义进行讨论与细化。第二部分将空间约束施加于输入与输出。多项式算法通常假设多项式采用稠密表示，即存储完整的系数列表。与之相对，我们处理的是稀疏多项式，其中大多数系数为零。特别地，我们描述了首个用于稀疏插值的拟线性算法，该算法在稀疏场景中发挥类似于快速傅里叶变换的作用。此外，我们还探讨了涉及稀疏多项式可整除性与因式分解的计算困难问题。