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|Author||Peter Bürgisser and Marek Karpinski and Thomas Lickteig|
|Abstract||We investigate the impact of randomization on the complexity of deciding membership in a (semi-)algebraic subset $X$ in $mathbb R^m$. Examples are exhibited, where allowing for a certain error probability in the answer of the algorithms the complexity of decision problems decreases. A randomized decision tree over $m$ will be defined as a pair $(T,u)$ where $u$ is a probability measure on some $mathbb R^n$ and $T$ is a decision tree over $m+n$. We prove a general lower bound on the average decision complexity for testing membership in an irreducible algebraic subset $X$ in $mathbb R^m$ and apply it to $k$-generic complete intersection of polynomials of the same degree, extending results in Buergisser, Lickteig, and Shub (1992), and Buergisser (1992). We also give applications to nongeneric cases, such as graphs of elementary symmetric functions, $mathrm Smathrm L(m,mathbb R)$, and determinant varieties, extending results in Lickteig (1990).|
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