Kevin Keating

Isohedral Polyomino Tiling of the Plane

Abstract. A polynomial time algorithm is given for deciding, for a given polyomino P , whether there exists an isohedral tiling of the Euclidean plane by isometric copies of P . The decidability question for general tilings by copies of a single polyomino, or even periodic tilings by copies of a single polyomino, remains open.

Ovals and Hyperovals in Desarguesian Nets

We determine the Desarguesian planes which hold r-nets with ovals and those which hold r-nets with hyperovals for every r≤7.

Enumeration of isomorphism classes of extensions of p-adic fields

Abstract Let Ω be an algebraic closure of Q p and let F be a finite extension of Q p contained in Ω . Given positive integers f and e, the number of extensions K/F contained in Ω with residue degree f and ramification index e was computed by Krasner. This paper is concerned with the number I (F,f,e) of F-isomorphism classes of such extensions. We determine I (F,f,e) completely when p 2 ∤e and get partial results when p 2 || e . When s is large, I ( Q p ,f,e) is equal to the number of isomorphism classes of finite commutative chain rings with residue field F p f , ramification index e, and length s.

Signed Tilings with Squares

LetTbe a bounded region in the Cartesian plane built from finitely many rectangles of the form [a1,,a2)×[b1,b2), witha1

Signed shape tilings of squares

Abstract Let T be a tile made up of finitely many rectangles whose corners have rational coordinates and whose sides are parallel to the coordinate axes. This paper gives necessary and sufficient conditions for a square to be tilable by finitely many Q -weighted tiles with the same shape as T, and necessary and sufficient conditions for a square to be tilable by finitely many Z -weighted tiles with the same shape as T. The main tool we use is a variant of F.W. Barness algebraic theory of brick packing, which converts tiling problems into problems in commutative algebra.