7 research outputs found

    On the Koiran-Skomra's question about Hessians

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    We give a negative answer to a question of Koiran and Skomra about Hessians, motivated by Kayal's algorithm for the equivalence problem to the Fermat polynomial. We conjecture that our counterexamples are the only ones. We also study a local version of their question.Comment: 14 p

    Black Box Absolute Reconstruction for Sums of Powers of Linear Forms

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    We study the decomposition of multivariate polynomials as sums of powers of linear forms. We give a randomized algorithm for the following problem: If a homogeneous polynomial f ? K[x_1. . .x_n] (where K ? ?) of degree d is given as a blackbox, decide whether it can be written as a linear combination of d-th powers of linearly independent complex linear forms. The main novel features of the algorithm are: - For d = 3, we improve by a factor of n on the running time from the algorithm in [Pascal Koiran and Mateusz Skomra, 2021]. The price to be paid for this improvement is that the algorithm now has two-sided error. - For d > 3, we provide the first randomized blackbox algorithm for this problem that runs in time poly(n,d) (in an algebraic model where only arithmetic operations and equality tests are allowed). Previous algorithms for this problem [Kayal, 2011] as well as most of the existing reconstruction algorithms for other classes appeal to a polynomial factorization subroutine. This requires extraction of complex polynomial roots at unit cost and in standard models such as the unit-cost RAM or the Turing machine this approach does not yield polynomial time algorithms. - For d > 3, when f has rational coefficients (i.e. K = ?), the running time of the blackbox algorithm is polynomial in n,d and the maximal bit size of any coefficient of f. This yields the first algorithm for this problem over ? with polynomial running time in the bit model of computation. These results are true even when we replace ? by ?. We view the problem as a tensor decomposition problem and use linear algebraic methods such as checking the simultaneous diagonalisability of the slices of a tensor. The number of such slices is exponential in d. But surprisingly, we show that after a random change of variables, computing just 3 special slices is enough. We also show that our approach can be extended to the computation of the actual decomposition. In forthcoming work we plan to extend these results to overcomplete decompositions, i.e., decompositions in more than n powers of linear forms

    Numerical algorithm and complexity analysis for diagonalization of multivariate homogeneous polynomials

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    We study the computational complexity of a criterion and an algorithm for diagonalization of multivariate homogeneous polynomials, that is, expressing them as sums of powers of independent linear forms. They are based on Harrison's center theory and only require solving linear and quadratic systems of equations. Detailed descriptions and computational complexity of each step of the algorithm are provided. The complexity analysis focuses on the impacts of problem sizes, including the number of variables and the degree of given polynomials. We show that this algorithm runs in polynomial time and validate it through numerical experiments. Other diagonalization algorithms are reviewed and compared in terms of complexity

    Efficient algorithms for some special cases of the polynomial equivalence problem

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