In one of Alon Amit's interesting answers on the Quora website, he mentioned Don Zagier's conjecture that the bivariate polynomial $x^7 + 3y^7$ may be injective on rational numbers, that is, no two distinct pairs of rational values of $(x,y)$ produce the same value of $x^7 + 3y^7$. Amit also referred to Bjorn Poonen's paper "Multivariable polynomial injections on rational numbers".
The choice of this particular polynomial by Zagier for this conjecture makes me curious: What about even simpler polynomials with smaller exponents? Presumably they are expected to be non-injective on rational numbers, if Zagier chose $x^7 + 3y^7$ in particular as a candidate to be injective.
But is there a proof that all polynomials of the forms
$$ ax^3 + by^3,\\ ax^3 + by^5,\\ ax^5 + by^5,\\ ax^3 + by^7,\\ ax^5 + by^7 $$
are non-injective on rational numbers?
Or at least, for small values of $a,b < 10$, are there known distinct pairs of rational values of $(x,y)$ showing that each such polynomial is non-injective?
Certain such polynomials have trivial or extremely simple examples showing that they are non-injective on rational numbers, such as $x^3 + 7y^3$: $(1,1)$ and $(2,0)$ produce identical values. Also, any polynomial of the form $x^n + 2y^n$, where $n$ is odd, has the trivial pair of rational values $(1,0)$ and $(-1,1)$ producing identical values. (Indeed any pair $(k,0)$ and $(-k, k)$ produce identical values of such polynomials). A less trivial example is $x^3 + 3y^3$, for which $(3,-1)$ and $(0,2)$ both produce the value of $24$.
But other such polynomials do not appear to have such simple non-injective solutions. For example, what is a pair of rational values of $(x,y)$ that produce identical values of $x^5 + 3y^5$? By the way, this is equivalent to finding a sum or difference of two fifth powers of integers that is exactly 3 times another such sum or difference of two fifth powers. Likewise, Zagier's conjecture that $x^7 + 3y^7$ is injective on rational numbers is equivalent to the claim that there does not exist any sum or difference of two seventh powers that is exactly 3 times another sum or difference of two seventh powers.
An example of a 5th degree polynomial with relatively small coefficients that I can show to be non-injective on rationals in a non-trivial way is $3x^5 + 22y^5$, for which $(1,1)$ and $(3,-2)$ both produce the value of $25$. Even better examples with very small coefficients are $x^5 + 4y^5$, for which $(1,4)$ and $(5,3)$ both produce the value of $4097$, and $x^5 + 8y^5$, for which $(1,4)$ and $(-7,5)$ both produce the value of $8193$. But it does not appear to be a simple matter to find such polynomials, and I imagine it would be quite difficult to show that all such 5th degree bivariate polynomials are non-injective on rationals.
The existence of the simple examples for 5th degree polynomials cited above is related to the fact that all fifth powers of integers are equivalent to 1, -1, or 0 modulo 11. This creates numerous small ratios of the many sums and/or differences of fifth powers which have 11 as a common factor: $2^5 + 1^5$, $3^5 - 1^5$, $4^5 - 3^5$, $5^5 - 1^5$, $5^5 - 4^5$, $7^5 + 1^5$, etc. Perhaps this is one reason that Zagier did not conjecture that a simple 5th degree bivariate polynomial is injective on rationals. It appears that the sums and differences of seventh powers do not share nearly as many common factors. Some of them have the common factor 29, since all seventh powers of integers are equivalent to 1, -1, 12, -12, or 0 modulo 29, and some have the common factor 43, since all seventh powers are equivalent to 1, -1, 6, -6, 7, -7, or 0 modulo 43. But these common factors do not appear to be dense enough to produce small ratios akin to those of the sums and differences of fifth powers.
Update: Robert Israel has found the following examples showing that certain additional 5th degree polynomials of the form $x^5 + by^5$ are non-injective on rational numbers:
$11^5 + 7^5 = 177858 = 6 * (8^5 - 5^5)$
$2698^5 + 1052^5 = 144246898755300000 = 16 * (1685^5 - 1355^5)$
$23^5 + 11^5 = 6597394 = 17 * (15^5 - 13^5)$
$59^5 + 17^5 = 716344156 = 19 * (39^5 - 35^5)$
$65^5 - 63^5 = 167854082 = 22 * (22^5 + 19^5)$
$131^5 - 116^5 = 17576073075 = 25 * (59^5 - 26^5)$
The lack of such examples with 5th powers up to $3000^5 = 243000000000000000$ (243 quadrillion) for $x^5 + 3y^5$, $x^5 + 5y^5$, $x^5 + 7y^5$, and multiples of $3, 5, 7$ except for $3*2 = 6$ and $5^2 = 25$ remains striking.
COMMENT.- Many elliptic curves, (when they have a rank $\ge1$), have their set of rational points, dense in the graph of the curve. So, for example $f: \mathbb Q\text { x }\mathbb Q\rightarrow\mathbb Q$ defined by $$f(x,y)=x^3+y^3$$ is largely non-injective, because, in particular, the preimage of $6, 12, 13,19$ and a lot of other integers is infinite.
Similar considerations should be maybe the reason of degree $7$ in Zagier's conjecture.