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Counting Real Roots in Polynomial-Time via Diophantine Approximation

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Abstract

Suppose \(A\!=\!\{a_1,\ldots ,a_{n+2}\}\!\subset \!\mathbb {Z}^n\) has cardinality \(n+2\), with all the coordinates of the \(a_j\) having absolute value at most d, and the \(a_j\) do not all lie in the same affine hyperplane. Suppose \(F\!=\!(f_1,\ldots ,f_n)\) is an \(n\times n\) polynomial system with generic integer coefficients at most H in absolute value, and A the union of the sets of exponent vectors of the \(f_i\). We give the first algorithm that, for any fixed n, counts exactly the number of real roots of F in time polynomial in \(\log (\textrm{d}H)\). We also discuss a number-theoretic hypothesis that would imply a further speed-up to time polynomial in n as well.

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Notes

  1. Roots yielding a Jacobian with less than full rank.

  2. Using Maple 2019 on a Dell XPS 13 laptop with an Intel core i7-5500u microprocessor, 8 Gb RAM, and a 256Gb solid state hard-drive, running Ubuntu 19.10. Maple code available on request.

  3. Via Kushnirenko’s Theorem [67, 91], Ioannis Emiris’ MixVol code [47], and a simple check that the underlying facial systems have no roots in \((\mathbb {C}^*)^5\).

  4. Even allowing degenerate isolated roots.

  5. [50] in fact proves a stronger theorem by using a weaker hypothesis that we will clarify below.

  6. See Sect. 2.4 for the definition of heights for algebraic numbers.

  7. Our stated bound assumes that we use an \(O(h\log h)\)-time algorithm for h-bit integer multiplication, e.g., [56].

  8. ...not to be confused with the circuits from complexity theory (which are layered directed graphs with specially labeled nodes having additional structure).

  9. L also happens to be increasing, with range \(\mathbb {R}\), on \((-\infty ,0)\) and (0.182377..., 0.332447...), and thus L has 2 more roots in \(\mathbb {R}^*\) that do not satisfy the necessary sign conditions.

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Acknowledgements

I thank Dan Bates and Jon Hauenstein for answering my questions on how Bertini handles polynomial systems of extremely high degree. I also thank Jan Verschelde for answering my questions on fine-tuning the options in PHCpack. Special thanks to Timo de Wolff for pointing out reference [35] and Alexander Barvinok for pointing out reference [85]. I also thank Weixun Deng, Alperen Ergür, and Grigoris Paouris for good company and inspirational conversations. I am also indebted to the referees for their detailed suggestions which greatly improved this paper. In particular, they encouraged me to include Theorem 1.6, and they pointed out Lemma 2.30 (which gives a simpler and sharper bound than I was using for the same purpose earlier), as well as a simplification of an earlier genericity condition for Theorem 1.1.

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  1. Department of Mathematics, Texas A & M University, TAMU 3368, College Station, TX, 77843-3368, USA

    J. Maurice Rojas

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Rojas, J.M. Counting Real Roots in Polynomial-Time via Diophantine Approximation. Found Comput Math 24, 639–681 (2024). https://doi.org/10.1007/s10208-022-09599-z

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