98 research outputs found
Random algebraic construction of extremal graphs
In this expository paper, we present a motivated construction of large graphs
not containing a given complete bipartite subgraph. The key insight is that the
algebraic constructions yield very non-smooth probability distributions.Comment: 8 page
Variety Evasive Sets
We give an explicit construction of a large subset of F^n, where F is a
finite field, that has small intersection with any affine variety of fixed
dimension and bounded degree. Our construction generalizes a recent result of
Dvir and Lovett (STOC 2012) who considered varieties of degree one (affine
subspaces).Comment: 13 page
Algorithms yield upper bounds in differential algebra
Consider an algorithm computing in a differential field with several
commuting derivations such that the only operations it performs with the
elements of the field are arithmetic operations, differentiation, and zero
testing. We show that, if the algorithm is guaranteed to terminate on every
input, then there is a computable upper bound for the size of the output of the
algorithm in terms of the input. We also generalize this to algorithms working
with models of good enough theories (including for example, difference fields).
We then apply this to differential algebraic geometry to show that there
exists a computable uniform upper bound for the number of components of any
variety defined by a system of polynomial PDEs. We then use this bound to show
the existence of a computable uniform upper bound for the elimination problem
in systems of polynomial PDEs with delays
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