History of Davenport-Stothers polynomial triples

 Title: Davenport-Stothers polynomial triples-Definition: A Davenport-Stothers polynomial triple of type $M$ is a triple $f,g,h\in\mathbb-  Q[t]$ with $h=f^3-g^2\ne0$, $\deg f=2M$, $\deg g=3M$ and $\deg h=M+1$. The half-degree-  $M$ and the tree label identify one equivalence class, and the part is one of its-  three polynomials.+Definition: A Davenport-Stothers polynomial triple of half-degree $M$ is a triple+  $f,g,h\in\mathbb Q[t]$ with $h=f^3-g^2\ne0$, $\deg f=2M$, $\deg g=3M$ and $\deg+  h=M+1$. Each row stores one representative of a listed equivalence class of such+  triples, and the part is one of its three polynomials. Parameters:   M:

     constraints: a positive integer with $\deg f=2M$     values:-      '1': $M=1$-      '2': $M=2$-      '3': $M=3$-      '4': $M=4$-      '5': $M=5$+      '1': '1'+      '2': '2'+      '3': '3'+      '4': '4'+      '5': '5'   tree:     type: Symbolic-    title: tree label-    constraints: a label for the equivalence class of the Hall dessin represented-      by the row+    title: equivalence class+    constraints: one of the equivalence classes of triples of the given half-degree     values:       unique: unique-      Birch: Birch+      Birch: Birch symmetric+    display: class   part:     type: Symbolic

       h: $h(t)$ Comments:-  comment-equivalence: Two triples are equivalent in the sense of CITE{Montanus2006}-    when constants $u,w\in\mathbb C^*$ and $v\in\mathbb C$ carry $f_1,g_1,h_1$ to-    $w^2 f_1(ut+v)$, $w^3 g_1(ut+v)$ and $w^6 h_1(ut+v)$. The table stores one representative-    of each class printed here, with the variable renamed to $t$.-  comment-tree-label: For $M\leq4$, CITE{Montanus2006} states that the displayed triples-    are the only ones up to equivalence, so their tree label is `unique`. For $M=5$,-    the stored row is Birch's symmetric example.+  comment-bound: Davenport's polynomial bound says that if $\deg f=2M$ and $f^3\ne+    g^2$, then $\deg(f^3-g^2)\geq M+1$ CITE{Montanus2006}. The rows attain the bound,+    so $\deg h=M+1$ is the extremal case, the polynomial analogue of HREF{Good_examples_of_Hall's_conjecture}[Hall's+    conjecture].+  comment-equivalence: Triples $(f,g,h)$ and $(f_1,g_1,h_1)$ are equivalent in the+    sense of CITE{Montanus2006} if $f_1(t)=w^2f(ut+v)$, $g_1(t)=w^3g(ut+v)$ and $h_1(t)=w^6h(ut+v)$+    for some $u,w\in\mathbb C^*$ and $v\in\mathbb C$. The table stores, for each listed+    class, the representative printed in CITE{Montanus2006}, with the variable renamed+    to $t$. An equivalent representative can have different coefficients until this+    substitution is made.+  comment-hall-tree: For a triple, the rational function $f(t)^3/h(t)=1+g(t)^2/h(t)$+    is a Belyi map CITE{Montanus2006}. Its dessin has $2M$ vertices over $0$, each+    of valence $3$, $3M$ vertices over $1$, each of valence $2$, and $M+1$ faces.+    Montanus's Hall tree is obtained from this dessin by removing the markings and+    cutting off the faces; it is a plane tree with $M+1$ leaves and $M-1$ trivalent+    internal vertices.+  comment-class-count: Montanus gives one deformation class, hence one equivalence+    class of triples, for each $M\leq4$, and four for $M=5$ CITE{Montanus2006}. The+    stored $M=5$ class is the symmetric Birch example. Formulas:-  formula-defining-relation:-    name: defining relation-    text: $h(t)=f(t)^3-g(t)^2$.-  formula-belyi-map:-    name: Belyi map-    text: $f(t)^3/h(t)=1+g(t)^2/h(t)$ is the Belyi map attached to the Hall dessin-      of the triple CITE{SijslingVoight}.+  formula-defining-relation: $h(t)=f(t)^3-g(t)^2$.+  formula-belyi-map: The Belyi map attached to the Hall dessin of the triple is $f(t)^3/h(t)=1+g(t)^2/h(t)$+    CITE{Montanus2006}. Similar tables: - table: HREF{Good_examples_of_Hall's_conjecture}[Good examples of Hall's conjecture]

 - table: HREF{Known_solutions_of_the_Fermat-Catalan_equation}[Known solutions of the     Fermat-Catalan equation]-  relation: records another family where perfect powers nearly coincide+  relation: records coprime perfect powers whose sum is a perfect power exactly - table: HREF{Counterexamples_to_Euler's_sum_of_powers_conjecture}[Counterexamples     to Euler's sum of powers conjecture]-  relation: records exact integer parts of another perfect-power identity+  relation: records integer solutions of $\sum a_i^k=b^k$ with fewer than $k$ terms Links:   WikiHall:

   rigour: exact   complete: 'no'-  complete-note: it holds the five examples printed in CITE{Montanus2006}; that source-    states uniqueness up to equivalence for the $M\leq4$ rows, while the $M=5$ row-    is Birch's symmetric example+  complete-note: it holds every equivalence class for $M\leq4$, and for $M=5$ only+    Birch's symmetric class among the four classes described by CITE{Montanus2006}   sources:   - CITE{Montanus2006}