We consider a random walk in a fixed Z environment composed of two point types: (q, 1 − q) and (p, 1 − p) for 1 2 < q < p. We study the expected hitting time at N for a given number k of p-drifts in the interval [1, N − 1], and find that this time is minimized asymptotically by equally spaced p-drifts.

Let k be a field and Q ∈ k[x 1 , . . . , x s ] a form (homogeneous polynomial) of degreeWhen k is algebraically closed, this rank is essentially equivalent to the codimension in k s of the singular locus of the variety defined by Q, known also as the Birch rank of Q. When k is a number field, a finite field or a function field, we give polynomial bounds for rk k (Q) in terms of rkk (Q) where k is the algebraic closure of k. Prior to this work no such bound (even ineffective) was known for d > 4. This result has immediate consequences for counting integer points (when k is a number field) or prime points (when k = Q) of the variety {Q = 0} assuming rk k (Q) is large.