Some algorithms for calculating a Groebner basis are optimized for calculating with coefficients in a finite field. Having determined the basis over a finite field, I'd like to understand what implications about the basis over the rational numbers can be made (and vice versa). I'm interested in implications that can be proven true as well as "heuristics", those implications that may only be true with some sense of high probability.
So given a set $P$ of multi-variate polynomials with integer coefficients, let us consider the ideal generated by this set of polynomials when considering the polynomial ring to be over the rational numbers, or a finite field $\mathbb{Z}/p\mathbb{Z}$.
I want to say $p$ is "large" in some sense, but I'm not sure what sense matters. Let $p$ be larger than the number of polynomials in $P$, more than the number of terms in any polynomial in $P$, and larger than the magnitude of any coefficient of any term in any polynomial in $P$.
Given some fixed monomial order, let's call the finite field Groebner basis $I_p$, and the rational Groebner basis $I_Q$.
Also let $J(I_p)$ be a set of polynomials over the rational numbers "converted" from $I_p$, by writing the polynomials in $I_p$ with coefficients converted to the minimum magnitude integer, and then reconsidered as a set of polynomials over the rational numbers.
Here are the kind of relationships I'm interested in knowing if they are true, or at least "high probability" (in the sense of randomly selecting a prime p in some range?).
- $J(I_p)$ is a Grobener basis
- $I_p$ is just $I_Q$ with the coefficients rewritten mod $p$.
- If $I_p=\{ 1 \}$ then $I_Q=\{ 1 \}$
- If $I_p\neq\{ 1 \}$ then $I_Q\neq \{ 1 \}$
- If $I_Q=\{ 1 \}$ then $I_p=\{ 1 \}$
- If $I_Q\neq\{ 1 \}$ then $I_p\neq \{ 1 \}$
If we imagined working through Buchberger's algorithm to compute a basis in the rational case (and where we only divide out common divisors of the coefficients, so as to leave the coefficients of the polynomials always integers, possibly making the leading term have a coefficient other than 1), there could be an S-polynomial that has a leading term with a coefficient a non-zero multiple of $p$ and so would look like it cancelled when calculating the basis for $I_p$ but not $I_Q$. For example, consider if the algorithm found the the constant $p$ was in the ideal. This would collapse $I_Q=\{ 1 \}$, but not $I_p$. Or maybe the algorithm finds $px+1$ was in the ideal. This would collapse $I_p=\{ 1 \}$, but not $I_Q$. So it seems like $I_p$ could be larger or smaller than $I_Q$, so I think any of those relationships could be false if cancellations dependent on the value of $p$ can occur during Buchberger's algorithm.
If cancellation dependent on the value of $p$ never occurs during the algorithm, then I think all of those relationships would be true. For a large $p$, it feels like these cancellations should be rare, so in the sense of selecting random $p$, I think these relationships should be "usually" true. Since the algorithm is finite, there exists some $n$ such that any coefficient $k$ which appears during calculation of the basis over the rationals will be between $-n/2 < k < n/2$. Then there exists some finite limit $n$, such that if $p > n$, we could know the relationships are true.
Is my understanding up to here correct?
If not, then where did my thinking go wrong is the real question.
Assuming I got at least the broad strokes correct, my question is then:
Is there a known constructive limit for $p$ that allows concluding the above relationships? For instance can this limit for $p$ be written in terms of the number of generating polynomials, the max number of terms, and the max magnitude of the coefficients?