580 research outputs found
Comparison of symbolic and ordinary powers of ideals
In this paper we generalize the theorem of Ein-Lazarsfeld-Smith (concerning
the behavior of symbolic powers of prime ideals in regular rings finitely
generated over a field of characteristic 0) to arbitrary regular rings
containing a field. The basic theorem states that in such rings, if P is a
prime ideal of height c, then for all n, the symbolic (cn)th power of P is
contained in the nth power of P. Results are also given in the non-regular
case: one must correct by a power of the Jacobian ideal in rings where the
Jacobian ideal is defined
Quantum Query Algorithms are Completely Bounded Forms
We prove a characterization of -query quantum algorithms in terms of the
unit ball of a space of degree- polynomials. Based on this, we obtain a
refined notion of approximate polynomial degree that equals the quantum query
complexity, answering a question of Aaronson et al. (CCC'16). Our proof is
based on a fundamental result of Christensen and Sinclair (J. Funct. Anal.,
1987) that generalizes the well-known Stinespring representation for quantum
channels to multilinear forms. Using our characterization, we show that many
polynomials of degree four are far from those coming from two-query quantum
algorithms. We also give a simple and short proof of one of the results of
Aaronson et al. showing an equivalence between one-query quantum algorithms and
bounded quadratic polynomials.Comment: 24 pages, 3 figures. v2: 27 pages, minor changes in response to
referee comment
- …