## On the efficiency of effective Nullstellensätze.(English)Zbl 0824.68051

Summary: Let $$k$$ be an infinite and perfect field, $$x_ 1, \dots, x_ n$$ indeterminates over $$k$$ and let $$f_ 1, \dots, f_ s$$ be polynomials in $$k[x_ 1, \dots, x_ n]$$ of degree bounded by a given number $$d$$, which satisfies $$d \geq n$$. We prove an effective affine Nullstellensatz of the following particular form: For arbitrary given parameters $$d,s,n$$ there exists a probabilistic (randomized) arithmetic network over $$k$$ of size $$s^{O (1)}$$ $$d^{O (n)}$$ and depth $$O(n^ 4 \log^ 2 sd)$$ solving the following task: It decides whether the ideal generated by $$f_ 1, \dots, f_ s$$ in $$k[x_ 1, \dots, x_ n]$$ is trivial and, if this is the case, it produces a straight-line program of size $$s^{O (1)}$$ $$d^{O (n)}$$ and depth $$O (n^ 4 \log^ 2 sd)$$ in the function field $$k(x_ 1, \dots, x_ n)$$ which computes polynomials $$p_ 1, \dots, p_ s$$ of $$k[x_ 1, \dots, x_ n]$$ of degree $$d^{O (n^ 2)}$$ satisfying $$1 = \sum_{1 \leq j \leq s} p_ j f_ j$$.

### MSC:

 68Q25 Analysis of algorithms and problem complexity
Full Text:

### References:

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