I am having difficulty understanding (also because of my native language) the expression that is widely used in mathematics:
"Uniquely Determined".
For example:
$1)$ Cat theorem asserts that the color of the cat $C$ is uniquely determined by its breed $b(C)$.
In this example $ 1) $ it is correct to understand that each breed of cat determines a single color? That is, cats of the race $ X $ can only be, say, blue and never of another color?
And the reciprocal, is it also true? That is, if we have a green cat, can we say that it can only be (solely) a single specific breed?
An example, minus "Alice in Wonderland", the following example is really what interests me:
$2)$"(...) The previously given forms of Torelli's theorem follow at onlce from the remark that the symmetric product $(X)^{(g-1)}$ is birationally equivalent to the canonical $\Theta$-divisor on $J(X)$, and the fact, proved by A. Weil, that the canonical polarization determines uniquely the $\Theta$-divisor."
In this example $2)$ it is correct to state that: the canonical polarization of $J(X)$ determines a single $\Theta$-divisor? Is it also reciprocal? That is, given a $\Theta$-divisor to it is associated a single polarization?
Thank You!
To say that $A$ is uniquely determined by $B$ is just to say that $A$ is a function of $B$. Note that if $A$ is uniquely determined by $B$, the converse need not follow. (It follows only when that function is invertible.)
For example, the sum of the squares of the eigenvalues of a $2\times 2$ matrix is uniquely determined by the determinant and trace, but the converse is not true.