Computing Higher Polynomial Discriminants
Computing Higher Polynomial Discriminants
复制标题
DOI：
10.1145/3452143.3465543
复制
发表时间：
2021-07
期刊：
影响因子：
--
通讯作者：
E. Kaltofen
中科院分区：
文献类型：
--
作者：
E. Kaltofen
在https://arxiv.org/abs/1609.00840（另请参见https://doi.org/10.1007/s11425-018-1594-2），2016年的Dongming Wang和Jing Yang在2016年提出了问题。多项式f（x）=（x-α1）…（x-αn），Δ3（f）=π（（αi+αj-αj-αk-αℓ）（αi-α-y-αj+αk-αℓ）（ αi-αJ-αK+αℓ））1≤<j <k <k <ℓn，从f的系数中，Δ3是αi中的对称多项式。 $ 2的$ 2的平均值是2个迭代的迭代式计算。第301--320页（1990）]计算其Squareroot。
In https://arxiv.org/abs/1609.00840 (see also https://doi.org/10.1007/s11425-018-1594-2), Dongming Wang and Jing Yang in 2016 have posed the problem how to compute the "third'' discriminant of a polynomial f(x) = (x-α1)…(x-αn), δ3(f) = Π ((αi+αj-αk-αℓ) (αi-αj+αk-αℓ)(αi-αj-αk+αℓ))1≤<j<k<ℓ≤n from the coefficients of f; note that δ3 is a symmetric polynomial in the αi. For complex roots, δ3(f) = 0 if the mid-point (average) of $2$ roots is equal the mid-point of another 2 roots. Iterated resultant computations yield the square of the third discriminant. We apply a symbolic homotopy by Kaltofen and Trager [JSC, vol. 9, nr. 3, pp. 301--320 (1990)] to compute its squareroot. Our algorithm uses polynomially many coefficient field operations in the degree of f.