论文标题
概括Tran的猜想
Generalizing Tran's Conjecture
论文作者
论文摘要
Khang Tran [6]的猜想声称，对于任意一对多项式$ a（z）$和$ b（z）$，每个多项式中的每个零元中的每个零零，$ \ {p_n（z）\} _ {n = 1} $$ p_n（z）+b（z）p_ {n-1}（z）+a（z）p_ {n-k}（z）= 0 $$具有标准初始条件$ p_0（z）= 1 $，$ p _ { - p _ { - 1}（z）（z）= \ dots = \ dots = p _ { - k+1} $ a（z）= 0 $ a（z） （semi） - algebraic曲线$ \ MATHCAL C \ subset \ MathBb {C} $由$ \ im \ left（\ frac {b^k（z）} {a（z）} {a（z）} \ right）= 0 \ quad \ quad {\ quad { \ left（\ frac {b^k（z）} {a（z）} \ right）\ le \ le \ frac {k^k} {（k-1）^{k-1}}。 $$ p_n（z）+b（z）p_ {n- \ ell}（z）+a（z）p_ {n-k}（z）（z）= 0，$$ 使用coprime $ k $和$ \ ell $以及与上述相同的标准初始条件，$ p_n（z）$的每个根都不是$ a（z）b（z）b（z）$的零零，属于真实的代数曲线$ \ mathcal c _ {\ ell，k}。 \ left（\ frac {b^k（z）} {a^\ ell（z）} \ right）= 0。$$
A conjecture of Khang Tran [6] claims that for an arbitrary pair of polynomials $A(z)$ and $B(z)$, every zero of every polynomial in the sequence $\{P_n(z)\}_{n=1}^\infty$ satisfying the three-term recurrence relation of length $k$ $$P_n(z)+B(z)P_{n-1}(z)+A(z)P_{n-k}(z)=0 $$ with the standard initial conditions $P_0(z)=1$, $P_{-1}(z)=\dots=P_{-k+1}(z)=0$ which is not a zero of $A(z)$ lies on the real (semi)-algebraic curve $\mathcal C \subset \mathbb {C}$ given by $$\Im \left(\frac{B^k(z)}{A(z)}\right)=0\quad {\rm and}\quad 0\le (-1)^k\Re \left(\frac{B^k(z)}{A(z)}\right)\le \frac{k^k}{(k-1)^{k-1}}.$$ In this short note, we show that for the recurrence relation (generalizing the latter recurrence of Tran) given by $$P_n(z)+B(z)P_{n-\ell}(z)+A(z)P_{n-k}(z)=0, $$ with coprime $k$ and $\ell$ and the same standard initial conditions as above, every root of $P_n(z)$ which is not a zero of $A(z)B(z)$ belongs to the real algebraic curve $\mathcal C_{\ell,k}$ given by $$\Im \left(\frac{B^k(z)}{A^\ell(z)}\right)=0.$$
