Projective line

Projective line

Geometry

Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006

If V is a two-dimensional vector space over the field of complex numbers, then the projective line P(V) can be realized as the set of all homogeneous coordinates [x: y] where x and y are complex numbers, not both zero. By identifying [x: y] with the complex number x/y if y is nonzero, and with ∞ if y = 0, we can identify P(V) with the extended complex plane studied in courses on complex analysis. Note also that the complex-linear transformation of V given by a nonsingular matrix [abcd] represents the mapping of the extended plane given by z↦az+bcz+d; these are the linear fractional transformations studied in such a course.

Minimality criteria for convergent power series over Z_p and rational maps with good reduction on the projective line over Q_p

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Journal Information

Published in Dynamical Systems, 2022

Sangtae Jeong, Dohyun Ko, Yongjae Kwon, Youngwoo Kwon

Let be the projective line over a finite field of p elements. The reduction induces a transformation from into itself. Throughout, denotes the kth iteration of ϕ. We are now ready to state the minimality criterion of from Fan et al.

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Geometry

Minimality criteria for convergent power series over Z p and rational maps with good reduction on the projective line over Q p

Minimality criteria for convergent power series over Z_p and rational maps with good reduction on the projective line over Q_p