Let's consider two non-singular integer matrices $A,B \in\mathbb{Z}^{n\times n}$. I want a test to check if $A\times B^{-1}$ is integral (or no denominators). I am referring the unimodular certification paper by Storjohann. The inverse of a matrix $M$ can be written as a $x$-adic expansion: $$M^{-1}=c_0+c_1x+ c_2x^2+\cdots $$ If the matrix $M$ is unimodular, the $x$-adic expansion is finite and Storjohann provides a fast algorithm to check this. My approach was: I computed few $x$-adic terms of $B^{-1}$. Say: $B^{-1}=b_0+b_1x+\cdots$. I can write matrix $A$ as a finite $x$-adic expansion. Let $A=a_0+a_1x+\cdots +a_m+x^m$. Similarly, if $A\times B^{-1}$ is integral, I should be able to write it as a finite $x$-adic expansion. Hence, I checked if $a_0 \times b_i$ (or $a_j\times b_i$) becomes zero at some point. I test for some examples (for matrices over Number fields), but it didn't work. Is there any problem with the theory? or Can someone suggest me a way to certify integrality of the product $A\times B^{-1}$.

Let's consider two non-singular integer matrices $A,B \in\mathbb{Z}^{n\times n}$. I want a test to check if $A\times B^{-1}$ is integral (or no denominators). I am referring the unimodular certification paper by Storjohann. The inverse of a matrix $M$ can be written as a $x$-adic expansion: $$M^{-1}=c_0+c_1x+ c_2x^2+\cdots $$ If the matrix $M$ is unimodular, the $x$-adic expansion is finite and Storjohann provides a fast algorithm to check this. (Here $x\in \mathbb{Z}_{>2}$ is relatively prime to determinant of $M$). My approach was: I computed few $x$-adic terms of $B^{-1}$. Say: $B^{-1}=b_0+b_1x+\cdots$. I can write matrix $A$ as a finite $x$-adic expansion. Let $A=a_0+a_1x+\cdots +a_m+x^m$. Similarly, if $A\times B^{-1}$ is integral, I should be able to write it as a finite $x$-adic expansion. Hence, I checked if $a_0 \times b_i$ (or $a_j\times b_i$) becomes zero at some point. I test for some examples (for matrices over Number fields), but it didn't work. Is there any problem with the theory? or Can someone suggest me a way to certify integrality of the product $A\times B^{-1}$.

Let's consider two non-singular integer matrices $A,B \in\mathbb{Z}^{n\times n}$. I want a test to check if $A\times B^{-1}$ is integral (or no denominators). I am referring the unimodular certification paper by Storjohann. The inverse of a matrix $M$ can be written as a $x$-adic expansion: $$M^{-1}=c_0+c_1x+ c_2x^2+\cdots $$ If the matrix $M$ is unimodular, the $x$-adic expansion is finite and Storjohann provides a fast algorithm to check this. My approach was: I computed few $x$-adic terms of $B^{-1}$. Say: $B^{-1}=b_0+b_1x+\cdots$. I can write matrix $A$ as a finite $x$-adic expansion. Let $A=a_0+a_1x+\cdots +a_m+x^m$. Similarly, if $A\times B^{-1}$ is integral, I should be able to write it as a finite $x$-adic expansion. Hence, I checked if $a_0 \times b_i$ (or $a_j\times b_i$) becomes zero at some point. I test for some examples (for matrices over Number fields), but it didn't work. Is there any problem with the theory? or Can someone suggest me a way to certify integrality of the product $A\times B^{-1}$. Thank you in advance.

Let's consider two non-singular integer matrices $A,B \in\mathbb{Z}^{n\times n}$. I want a test to check if $A\times B^{-1}$ is integral (or no denominators). I am referring the unimodular certification paper by Storjohann. The inverse of a matrix $M$ can be written as a $x$-adic expansion: $$M^{-1}=c_0+c_1x+ c_2x^2+\cdots $$ If the matrix $M$ is unimodular, the $x$-adic expansion is finite and Storjohann provides a fast algorithm to check this. My approach was: I computed few $x$-adic terms of $B^{-1}$. Say: $B^{-1}=b_0+b_1x+\cdots$. I can write matrix $A$ as a finite $x$-adic expansion. Let $A=a_0+a_1x+\cdots +a_m+x^m$. Similarly, if $A\times B^{-1}$ is integral, I should be able to write it as a finite $x$-adic expansion. Hence, I checked if $a_0 \times b_i$ (or $a_j\times b_i$) becomes zero at some point. I test for some examples (for matrices over Number fields), but it didn't work. Is there any problem with the theory? or Can someone suggest me a way to certify integrality of the product $A\times B^{-1}$.