
TL;DR
This paper introduces the concept of 'bias' into Grothendieck's dessins theory, creating new algebraic and combinatorial structures that enhance the understanding of Galois actions and algebraic numbers.
Contribution
It adds a new 'bias' concept to dessins, enabling the construction of algebraic number sequences and Galois-invariant lattices, expanding the theoretical framework.
Findings
Defined a new biased lattice structure of dessins
Constructed algebraic number sequences from biased trees
Identified Galois invariants within the biased framework
Abstract
Grothendieck's theory of dessins provides a bridge between algebraic numbers and combinatorics. This paper adds a new concept, called 'bias', to the bridge. This produces: (i) from a biased plane tree the construction of a sequence of algebraic numbers, and (ii) a Galois invariant lattice structure on the set of biased dessins. Bias brings these benefits by (i) using individual polynomials instead of equivalence classes of polynomials, and (ii) applying properties of covering spaces and the fundamental group. The new features give new opportunities. At the 2014 SIGMAP conference the author spoke [1] on 'The decorated lattice of biased dessins'. This decorated lattice is combinatorially defined, and its automorphism group contains the absolute Galois group , perhaps as an index 6 subgroup. This paper defines new families of invariants of dessins, although they…
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Taxonomy
TopicsEducation, sociology, and vocational training
