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On the efficiency of effective Nullstellensätze

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Abstract

Letk be an infinite and perfect field,x 1, ...,x n indeterminates overk and letf 1, ...,f s be polynomials ink[x 1, ...,x n ] of degree bounded by a given numberd, which satisfiesdn. We prove an effective affine Nullstellensatz of the following particular form:

For arbitrary given parametersd, s, n there exists a probabilistic (randomized) arithmetic network overk of sizes O(1) d O(n) and depthO(n 4log2 sd) solving the following task:

It decides whether the ideal generated byf 1, ...,f s ink[x 1, ...,x n ] is trivial and, if this is the case, it produces a straight-line program of sizes O(1) d O(n) and depthO(n 4log2 sd) in the function fieldk(x 1, ...,x n ) which computes polynomialsp 1, ...,p s ofk[x 1, ...,x n ] of degree\(d^{O(n^2 )} \) satisfying

$$1 = \mathop \sum \limits_{1 \leqslant j \leqslant s} p_j f_j .$$

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Giusti, M., Heintz, J. & Sabia, J. On the efficiency of effective Nullstellensätze. Comput Complexity 3, 56–95 (1993). https://doi.org/10.1007/BF01200407

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