This paper studies ‘pro-excision’ for the $K$-theory of one-dimensional, usually semi-local, rings and its various applications. In particular, we prove Geller’s conjecture for equal characteristic rings over a perfect field of finite characteristic, give results towards Geller’s conjecture in mixed characteristic, and we establish various finiteness results for the $K$-groups of singularities, covering both orders in number fields and singular curves over finite fields.

This paper is concerned with non-trivial solvability in $p$-adic integers of systems of two and three additive forms. Assuming that the congruence equation $a x^k + b y^k + c z^k \equiv d \,(\mbox{mod}\,p)$ has a solution with $xyz \not\equiv 0\,(\mbox{mod}\,p)$ we have proved that any system of two additive forms of odd degree $k$ with at least $6 k + 1$ variables, and any system of three additive forms of odd degree $k$ with at least $14 k + 1$ variables always has non-trivial $p$-adic solutions, provided $p$ does not divide $k$. The assumption of the solubility of the congruence equation above is guaranteed for example if $p > k^4$.

In the particular case of degree $k = 5$ we have proved the following results. Any system of two additive forms with at least $n$ variables always has non-trivial $p$-adic solutions provided $n \geq 31$ and $p > 101$ or $n \geq 36$ and $p > 11$. Furthermore any system of three additive forms with at least $n$ variables always has non-trivial $p$-adic solutions provided $n \geq 61$ and $p > 101$ or $n \geq 71$ and $p > 11$.

We study dynamics of rational maps of degree at least 2 with coefficients in the field ${\open C}_{p}$, where p>1 is a fixed prime number. The main ingredient is to consider the action of rational maps in p-adic hyperbolic space, denoted ${\open H}_{p}$. Hyperbolic space ${\open H}_{p}$ is provided with a natural distance, for which it is connected and one dimensional (an ${\open R}$-tree). These advantages with respect to ${\open C}_{p}$ give new insight into dynamics. In this paper we prove the following results about periodic points; we give applications to the Fatou/Julia theory over ${\open C}_{p}$ and to ultrametric analysis in forthcoming papers. We prove that the existence of at least two nonrepelling periodic points implies the existence of infinitely many of them. This is in contrast with the complex setting where a rational map can have at most finitely many nonrepelling periodic points. On the other hand we prove that every rational map has a repelling fixed point, either in the projective line or in hyperbolic space. So the topological expansion of a rational map is detected by some fixed point.

We investigate the Chow groups of zero cycles of products of curves over a p-adic field by means of the Milnor K-groups of their Jacobians as introduced by Somekawa. We prove some finiteness results for CH0(X)/m for X a product of curves over a p-adic field.

Let p be a rational prime number. We refine Brauer‘s elementary diagonalisation argument to show that any system of r homogeneous polynomials of degree d, with rational coefficients, possesses a non-trivial p-adic solution provided only that the number of variables in this system exceeds $(rd^2)^2^(d-1)$. This conclusion improves on earlier results of Leep and Schmidt, and of Schmidt. The methods extend to provide analogous conclusions in field extensions of Q, and in purely imaginary extensions of Q. We also discuss lower bounds for the number of variables required to guarantee local solubility.