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polynomial in the number of bits in M
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Comment: You seem to be looking for best simultaneous diophantine approximations of a certain kind. In the original LLL article, "Factoring Polynomials with Rational Coefficients", the authors (Lenstra, Lenstra and Lovasz) describe how to find approximate solutions (in polynomial time) to very similar problems at the end of section 1.

Comment: You seem to be looking for best simultaneous diophantine approximations of a certain kind. In the original LLL article, "Factoring Polynomials with Rational Coefficients", the authors (Lenstra, Lenstra and Lovasz) describe how to find approximate solutions to very similar problems at the end of section 1.

Comment: You seem to be looking for best simultaneous diophantine approximations of a certain kind. In the original LLL article, "Factoring Polynomials with Rational Coefficients", the authors (Lenstra, Lenstra and Lovasz) describe how to find approximate solutions (in polynomial time) to very similar problems at the end of section 1.

Trying to answer a query from the op.
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Comment: You seem to be looking for best simultaneous diophantine approximations of a certain kind. TheIn the original LLL article describes, "Factoring Polynomials with Rational Coefficients", the authors (Lenstra, Lenstra and Lovasz) describe how to find approximate solutions to very similar problems at the end of section 1.

Comment: You seem to be looking for best simultaneous diophantine approximations of a certain kind. The original LLL article describes how to find approximate solutions to very similar problems.

Comment: You seem to be looking for best simultaneous diophantine approximations of a certain kind. In the original LLL article, "Factoring Polynomials with Rational Coefficients", the authors (Lenstra, Lenstra and Lovasz) describe how to find approximate solutions to very similar problems at the end of section 1.

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Comment: You seem to be looking for best simultaneous diophantine approximations of a certain kind. The original LLL article describes how to find approximate solutions to very similar problems.