Sunday, July 08, 2018

The determinant of an elliptic, Sylvesteresque matrix

My second determinant project with Christian Krattenthaler.

The determinant of the Sylvester matrix corresponding to the polynomials
\[
x^2+2s_1x+s_1^2 = (x+s_1)^2
\]
and
\[
x^3+3 s_2 x^2 +3s_2^2 x + s_2^3 = (x+s_2)^3
\]
is given by
\[
\det
\begin{pmatrix}
1 & 2s_1 & s_1^2 & 0 & 0\\
0 & 1& 2s_1 & s_1^2 & 0\\
0 & 0 &1 & 2s_1 & s_1^2 \\
1 & 3s_2 &  3s_2^2 & s_2^3 & 0\\
0& 1 & 3s_2 &  3s_2^2 & s_2^3 \\
\end{pmatrix}
= (s_1-s_2)^6.
\]

The determinant is $0$ when $s_1$ and $s_2$ are both $1$. In general, if the determinant of a Sylvester matrix is $0$, then this indicates that the two polynomials have a common root.

Here is an abstract of our paper.

We evaluate the determinant of a matrix whose entries are  elliptic hypergeometric terms and whose form is reminiscent of Sylvester matrices. In particular, it generalizes the determinant evaluation above. A hypergeometric determinant evaluation of a matrix of this type has appeared in the context of approximation theory, in the work of  Feng, Krattenthaler and Xu.  Our determinant evaluation is an elliptic extension of their evaluation, which has two additional parameters (in addition to the base $q$ and nome $p$ found in elliptic hypergeometric terms).  We also extend the evaluation to a formula transforming an elliptic determinant into a multiple of another elliptic determinant. This transformation has two further parameters. The proofs of the determinant evaluation and the transformation formula require an elliptic determinant lemma due to Warnaar, and the application of two $C_n$ elliptic formulas that extend Frenkel and Turaev's $_{10}V_9$ summation formula and $_{12}V_{11}$ transformation formula, results due to Warnaar, Rosengren, Rains, and Coskun and Gustafson.   

This paper has been published in Sigma. Here is a link:
The determinant of an elliptic Sylvesteresque matrix (with Christian Krattenthaler), SIGMA, 14 (2018), 052, 15pp.

I presented this paper in Combinatory Analysis 2018, a conference in honor of George Andrews' 80th birthday conference. Here is a picture from Andrews' talk. (The picture inside the picture is of Freeman J. Dyson.)



Next I expect to present the same paper in a Summer Research Institute on $q$-series in the University of Tianjin, China.

A long version (with lots of background information) was presented in our "Arbeitsgemeinschaft "Diskrete Mathematik" (working group in Discrete Mathematics) Seminar, TU-Wien and Uni Wien, on Tuesday, June 5, 2018. 


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