On solutions of polynomial Pell's equations and the continued fraction factorization algorithm

Abel在寻找可用初等函数表示的椭圆积分的过程中，首先对多项式Pell方程进行了研究。这是在有理数领域完成的。当我们将多项式Pell方程的解限制为整数系数多项式时，已知的结果只适用于一元四次多项式。我们在2003年证明了多项式Pell方程对于D = a ^2+2C和a /C∈Q[x]具有非平凡整数系数多项式解的充要条件。在本研究中，我们与webb教授合作，利用√<D>的连分式展开式的周期，研究了与部分商产生的椭圆曲线上有理点有关的多项式Pell方程。我们还通过观察小周期研究了多项式佩尔方程。对于单四次多项式D，我们能够证明不存在周期为3的连分数展开。对于单多项式D，我们能够证明连分式展开式的周期值是偶的当且仅当多项式Pell方程X^2- dy ^2 = 1有非平凡解。对于一元四次多项式D，我们能够证明多项式Pell方程X^2- dy ^2 = 1在Q[X]中有非平凡解当且仅当连分式展开式的周期值为2,4,6,8,10,14,18,22。

Study on the polynomial Pell's equation was first done by Abel in the connection of finding an elliptic integral which can be expressed using an elementary function. This was done in the field of rational numbers.When we restrict solutions of the polynomial Pell's equation to be integer coefficient polynomial, the known result is only for a monic quartic polynomial. We have shown in 2003, a necessary and sufficient condition for the polynomial Pell's equation has a nontrivial integer coefficient polynomial solution for D = A^2+2C and A/C∈Q[x].In this research, collaborating with Prof.Webb, we have studied the polynomial Pell's equation using the period of continued fraction expansions of √<D> in the connection with rational points on the elliptic curve arising from the partial quotients. We also have studied the polynomial Pell's equation by looking at the small periods.For D a monic quartic polynomial, we are able to show that there is no period 3 continued fraction expansion.For D a monic polynomial, we are able to show that the values of period of continued fraction expansions are even if and only if the polynomial Pell's equation X^2-DY^2 = 1 has a nontrivial solution.For D a monic quartic polynomial, we are able to show that the polynomial Pell's equation X^2-DY^2 = 1 has a nontrivial solution in Q[x] if and only if the values of the period of continued fraction expansions are 2,4,6,8,10,14,18,22.