Honestly I do not understand why Lenstra's MILP is in $P$ if the number of integer dimensions is fixed.

Here is what Lenstra says in 'http://people.csail.mit.edu/rrw/presentations/Lenstra81.pdf' in section 5. Mixed integer linear programming.

"At a certain point in the algorithm we have to decide whether a given vector $y\in\mathbb R^n$ belongs to $τΚ$. This can be done by solving a linear programming problem with $k — n$ variables".

Here $n$ is number of integer variables and $k-n$ is number of real variables that are not restricted to be integers.

By 'At a certain point in the algorithm' I take it as that first the algorithm finds an integer point in the first $n$ coordinates and by 'we have to decide whether a given vector $y\in\mathbb R^n$ belongs to $τΚ$' I take it that at this set coordinate of first $n$ coordinates it tests if the remaining $k-n$ real variables pass Khachiyan's test.

Is this right or close to being right?

Assume it is right then I claim this cannot be in polynomial time.

Consider $n=1$ and $0\leq x_1\leq M$ where $M$ has $m$ bits (where $m$ is number of constraints in the MILP which is not fixed). There are $O(2^m)$ integer points. I can easily cook up a linear program which has $k-1$ variables but has no feasible solutions. Then Lenstra's algorithm has to test for real constraints at each and every integer point and then return no. This has exponential complexity.

If this is not what Lenstra does then what exactly does he do? What exactly does he mean by 'At a certain point in the algorithm'?

Honestly I do not understand why Lenstra's MILP is in $P$ if the number of integer dimensions is fixed.

Here is what Lenstra says in 'http://people.csail.mit.edu/rrw/presentations/Lenstra81.pdf' in section 5. Mixed integer linear programming.

"At a certain point in the algorithm we have to decide whether a given vector $y\in\mathbb R^n$ belongs to $τΚ$. This can be done by solving a linear programming problem with $k — n$ variables".

Here $n$ is number of integer variables and $k-n$ is number of real variables that are not restricted to be integers.

By 'At a certain point in the algorithm' I take it as that first the algorithm finds an integer point in the first $n$ coordinates and by 'we have to decide whether a given vector $y\in\mathbb R^n$ belongs to $τΚ$' I take it that at this set coordinate of first $n$ coordinates it tests if the remaining $k-n$ real variables pass Khachiyan's test.

Is this right or close to being right? If this is not what Lenstra does then what exactly does he do and what exactly does he mean by 'At a certain point in the algorithm'?