**Eupolars and their Bialternality Grid **

* Jean Ecalle *

**Abstract**

This monograph is almost entirely devoted to the flexion structure generated by a flexion unit \(\boldsymbol {\mathfrak {E}}\) or the conjugate unit \(\boldsymbol {\mathfrak {O}}\), with special emphasis on the polar specialization of the units (“eupolar structure”). (i) We first state and prove the main facts (some of them new) about the central pairs of bisymmetrals pal^{∙}/pil^{∙} and par^{∙}/pir^{∙} and their even/odd factors, by relating these to four remarkable series of alternals \(\{\boldsymbol {\mathfrak {re}}_{r}^{\bullet }\}\), \(\{\boldsymbol {\mathfrak {le}}_{r}^{\bullet }\}\), \(\{\boldsymbol {\mathfrak {he}}_{r}^{\bullet }\}\), \(\{\boldsymbol {\mathfrak {ke}}_{2r}^{\bullet }\}\), and that too in a way that treats the swappees pal^{∙} and pil^{∙} (resp. par^{∙} and pir^{∙}) as they should be treated, i.e., on a strictly equal footing. (ii) Next, we derive from the central bisymmetrals two series of bialternals, distinct yet partially (and rather mysteriously) related. (iii) Then, as a first step towards a complete description of the eupolar structure, we introduce the notion of bialternality grid and present some facts and conjectures suggested by our (still ongoing) computations. (iv) Lastly, two complementary sections have been added, to show which features of the eupolar structure survive, change form, or altogether disappear when one moves on to the next two cases in order of importance: eutrigonometric and polynomial.