# Generic reductions for in-place polynomial multiplication

## TL;DR

This paper develops generic methods to perform in-place polynomial multiplication, including full, middle, and short products, maintaining asymptotic time complexity and broadening the applicability of space-efficient algorithms.

## Contribution

It introduces a series of reductions that enable any linear-space polynomial multiplication algorithm to be adapted for in-place use, including variants producing partial results.

## Key findings

- In-place algorithms for full and short products match out-of-place complexity.
- In-place middle product algorithms incur only a logarithmic time overhead.
- Reductions apply broadly to linear-space multiplication algorithms.

## Abstract

The polynomial multiplication problem has attracted considerable attention since the early days of computer algebra, and several algorithms have been designed to achieve the best possible time complexity. More recently, efforts have been made to improve the space complexity, developing modified versions of a few specific algorithms to use no extra space while keeping the same asymptotic running time. In this work, we broaden the scope in two regards. First, we ask whether an arbitrary multiplication algorithm can be performed in-place generically. Second, we consider two important variants which produce only part of the result (and hence have less space to work with), the so-called middle and short products, and ask whether these operations can also be performed in-place. To answer both questions in (mostly) the affirmative, we provide a series of reductions starting with any linear-space multiplication algorithm. For full and short product algorithms these reductions yield in-place versions with the same asymptotic time complexity as the out-of-place version. For the middle product, the reduction incurs an extra logarithmic factor in the time complexity only when the algorithm is quasi-linear.