June 30 (Wednesday) 2021 16:00 - 18:00, Building 110, Room N103


▶Speaker : Dayoon Park (UNIST)

▶Title : An applicable representation of quadratic form to show representation of m-gonal form


▶Abstract : Since the smallest $m$-gonal number is $m-3$ except $0$ and $1$, in constructing an universal $m$-gonal form, escalating process to represent from $1$ to at least $m-4$ must occur. But the representability of every positive integer up to $m-4$ does not characterize the universality i.e., there are $m$-gonal forms which represent every positive integer up to $m-4$ but not universal. Then one may question that how far is it to an $m$-gonal form which represents every positive integer up to $m-4$ from an universal form? Firstly, we show that for any $m$-gonal form which represents every positive integer up to $m-4$, to complete a universal form one more extra unary piece would be enough for $m \ge 12$. Even though the representability of every positive integer up to $m-4$ is not enough to imply the universality in general, for some specific pairs of first coefficients, the representability of $m$-gonal form whose first coefficients are specific (kinds of very nice) is equivalent its universality. Secondly, we classify such the first five coefficients and see some applications to determine the optimal (i.e., minimal) rank of some specific type of $m$-gonal forms to be univesal. In order to show both of above two arguments, we adopt the arithmetic theory of quadratic form. Especially, the proof relies on observation of particular representation of specific type of binry quadratic form by a diagonal quadratic form.


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