On semiring complexity of Schur polynomials

Semiring complexity is the version of arithmetic circuit complexity that allows only two operations: addition and multiplication. We show that semiring complexity of a Schur polynomial {s_\lambda(x_1,\dots,x_k)} labeled by a partition {\lambda=(\lambda_1\ge\lambda_2\ge\cdots)} is bounded by {O(\log(\lambda_1))} provided the number of variables $k$ is fixed

Shadows of Newton Polytopes

We define the shadow complexity of a polytope P as the maximum number of vertices in a linear projection of P to the plane. We describe connections to algebraic complexity and to parametrized optimization. We also provide several basic examples and constructions, and develop tools for bounding shadow complexity

Integer Complexity, Addition Chains, and Well-Ordering.

In this dissertation we consider two notions of the "complexity" of a natural number, one being addition chain length, the other known as "integer complexity". The integer complexity of n, denoted ||n||, is the smallest number of 1's needed to write n using an arbitrary combination of addition and multiplication. It is known that n>=3log_3(n) for all n. We consider the difference delta(n):=||n||−3log_3(n), which we call the defect of n. We consider the set of all defects - the set D:={delta(n) : n>=1}. We show that, as a set of real numbers, D is well-ordered, with order type omega^omega; we also show the same for several variants of D. Moreover, we show that, for k>=1 a natural number, the intersection of D with [0, k) has order type omega^k. We also use the defect to prove stabilization results about ||n||. Specifically, for any n, there exists K=K(n) such that for k>=K, we have delta(3^k*n)=delta(3^K*n). We call K(n) the stabilization length of n. Finally, we provide a way of, given r>0, computing all numbers n with delta(n)=0 with k+l>0. In parallel to our results for integer complexity, we also consider addition chain length. An addition chain for n is a sequence (a_0,a_1,...,a_r) such that a_0=1, a_r=n, and, for any k with 1<=k<=r, there exist 0<=i,j<=k such that a_k=a_i+a_j; the number r is the length of the chain. The shortest length among addition chains for n, the addition chain length of n, is denoted l(n). The number l(n) is always at least log_2(n). We consider the difference delta^l(n):=l(n)-log_2(n), which we call the addition-chain defect of n, and the set of all addition-chain defects D^l := {delta^l(n) : n>=1}. We show that D is also a well-ordered set with order type omega^omega. We also use the defect to prove stabilization results about l(n); specifically, for any n, there exists K'=K'(n) such that for k>=K', we have delta^l(2^k*n)=delta^l(2^K'*n).PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/108986/1/haltman_1.pd