To find the polynomial equation in terms of $x$ and $y$ that represents the geometric shape formed by the given parametric curve, we can eliminate the parameter $t$ from the system of equations:

$x(t) = t + \frac{1}{t}, \quad y(t) = t - \frac{1}{t}$

Step 1: Solve for $t$ in terms of $x$ and $y$

First, let's add the equations for $x(t)$ and $y(t)$:

$x(t) + y(t) = \left(t + \frac{1}{t}\right) + \left(t - \frac{1}{t}\right)$

Simplifying this expression:

$x(t) + y(t) = 2t$

Thus, we can express $t$ as:

$t = \frac{x(t) + y(t)}{2}$

Step 2: Subtract the equations for $x(t)$ and $y(t)$

Next, let's subtract the equation for $y(t)$ from the equation for $x(t)$:

$x(t) - y(t) = \left(t + \frac{1}{t}\right) - \left(t - \frac{1}{t}\right)$

Simplifying:

$x(t) - y(t) = 2 \cdot \frac{1}{t}$

Thus, we can express $\frac{1}{t}$ as:

$\frac{1}{t} = \frac{x(t) - y(t)}{2}$

Step 3: Eliminate $t$

Now that we have expressions for $t$ and $\frac{1}{t}$ in terms of $x$ and $y$, we substitute them into one of the original equations. Let's use the equation for $x(t)$:

$x = t + \frac{1}{t}$

Substitute $t = \frac{x + y}{2}$ and $\frac{1}{t} = \frac{x - y}{2}$:

$x = \frac{x + y}{2} + \frac{x - y}{2}$

Simplify the right-hand side:

$x = \frac{(x + y) + (x - y)}{2} = \frac{2x}{2} = x$

Thus, both sides match, confirming that the parametric equations describe a consistent relation.

Step 4: Final Polynomial Equation

Through this process, we confirmed that the parametric equations don't immediately suggest a simple polynomial relationship between $x$ and $y$ as we hoped. However, more advanced methods (like converting into more complex forms) can be used.

Would you like further details on different methods for parametrizing relations like this?