On decomposition of sub-linearised polynomials

We give a detailed exposition of the theory of decompositions of linearised polynomials, using a well-known connection with skew-polynomial rings with zero derivative. It is known that there is a one-to-one correspondence between decompositions of linearised polynomials and sub-linearised polynomials. This correspondence leads to a formula for the number of indecomposable sub-linearised polynomials of given degree over a finite field. We also show how to extend existing factorisation algorithms over skew-polynomial rings to decompose sub-linearised polynomials without asymptotic cost.

Ecology of Leptocoris Hahn (Hemiptera: Rhopalidae) Soapberry Bugs in Australia

Soapberry bugs are worldwide seed predators of plants in the family Sapindaceae. Australian sapinds are diverse and widespread, consisting of about 200 native trees and shrubs. This flora also includes two introduced environmental weeds, plus cultivated lychee (Litchi chinensis Sonn.), longan (Dimocarpus longan Lour.) and rambutan (Nephelium lappaceum L.). Accordingly, Australian soapberry bugs may be significant in ecology, conservation and agriculture. Here we provide the first account of their ecology. We find five species of Leptocoris Hahn in Australia, and list sapinds that do and do not serve as reproductive hosts. From museum and field records we map the continental distributions of the insects and primary hosts. Frequency of occupation varies among host species, and the number of hosts varies among the insects. In addition, differences in body size and beak length are related to host use. For example, the long-beaked Leptocoris tagalicus Burmeister is highly polyphagous in eastern rainforests, where it occurs on at least 10 native and non-native hosts. It aggregates on hosts with immature fruit and commences feeding before fruits dehisce. Most of its continental range, however, matches that of a single dryland tree, Atalaya hemiglauca F. Muell., which has comparatively unprotected seeds. The taxon includes a smaller and shorter-beaked form that is closely associated with Atalaya, and appears to be taxonomically distinct. The other widespread soapberry bug is the endemic Leptocoris mitellatus Bergroth. It too is short-beaked, and colonises hosts phenologically later than L. tagalicus, as seeds become more accessible in open capsules. Continentally its distribution is more southerly and corresponds mainly to that of Alectryon oleifolius Desf. Among all host species, the non-native environmental weeds Cardiospermum L. and Koelreuteria Laxm. are most consistently attacked, principally by L. tagalicus. These recent host shifts have biocontrol implications. In contrast, the sapinds planted as fruit crops appear to be less frequently used at present and mainly by the longer-beaked species.