@MISC{Landau93anote, author = {Susan Landau}, title = {A Note on "Zippel Denesting"}, year = {1993} }

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Abstract

Radical simplification is an important part of symbolic computation systems. Zippel [7] gave a sufficient condition for a nested radical to be expressed in terms of radicals of lower nesting depth. We fill a lacuna in his proof, and show that his sufficient condition is also necessary. Previous work by Landau and Miller [4] leads to an algorithm for the problem. Ramanujan observed a number of curiosities amongst nested radicals: 4 s 3 + 2 4 p 5 3 \Gamma 2 4 p 5 = 4 p 5 + 1 4 p 5 \Gamma 1 q 3 p 28 \Gamma 3 p 27 = 1=3( 3 p 98 \Gamma 3 p 28 \Gamma 1) 3 r 5 q 32=5 \Gamma 5 q 27=5 = 5 q 1=25 + 5 q 3=25 \Gamma 5 q 9=25 Symbolic computation made such manipulations more than curiosities. For example: q 5 + 2 p 6 = p 2 + p 3 means that f1; p 2; p 3; p 6g is a basis for Q( q 5 + 2 p 6) over Q. The basis f1; p 2; p 3; p 6g is simple to manipulate. An important issue then, is the Supported by NSF grants DMS-8807202, and CCR-8802835. Pa...