On the Computation of Minimal Free Resolutions with Integer Coefficients

Abstract

Let $I=\langle f_1,\ldots ,f_s\rangle$ be an ideal of $R=\mathbb {Z}[x_1,\ldots ,x_n]$. We introduce in this paper the concept of $\mathbb {Z}-$ideal $\mathbb {Z}(I)$ of I which is a proper ideal of R and we propose a technique for computing a weak Gröbner basis for $\mathbb {Z}(I)$. This result is central and leads to the computation of a minimal free resolution for $\mathbb {Z}(I)$ as an $R-$module.

Appendices

A Appendix

Basic notions

Definition A.1

[4]. Let $R=\mathbb {Z}[x_1,\ldots ,x_n]$ and $F=R^r$ be an $R-$free module with canonical basis $e_1,\ldots ,e_r$ for some $r>1$.

• 1. A monomial in R is a multivariate polynomial of the form $x_1^{\alpha _1}\cdots x_n^{\alpha _n}$ where $\alpha _i\in \mathbb {N}$. We denote $x^{\alpha }=x_1^{\alpha _1}\cdots x_n^{\alpha _n}$ where $\alpha =(\alpha _1,\ldots ,\alpha _n)$. We denote by $\text {Mon}(x_1,\ldots ,x_n)$ the set of all monomials in R.

• 2. A monomial in F is a product of the form $x^{\alpha }e_i$ where $x^{\alpha }$ is a monomial in R and $e_i$ an element of the basis. We denote by $\text {Mon}(F)$ the set of all monomials in F.

• 3. Accordingly, a term in F is the product of a monomial in F with an element of $\mathbb {Z}$.

• 4. A term $c_1x^{\alpha }e_i$ divides a term $c_2x^{\beta }e_j$ if the following conditions are satisfied:

  • $i=j$;

  • $c_1$ divides $c_2$ in $\mathbb {Z}$;

  • $x^{\alpha }$ divides $x^{\beta }$ in R.

  We also say that a monomial $x^{\alpha }\in \text {Mon}(x_1,\ldots ,x_n)$ divides a monomial $x^{\beta }e_j\in \text {Mon}(F)$ if $x^{\alpha }$ divides $x^{\beta }$ and in this case we write $(\displaystyle \frac{x^{\beta }}{x^{\alpha }})e_j\in \text {Mon}(F)$.

• 5. Let $x^{\beta }e_i,x^{\beta }e_j\in \text {Mon}(F)$, we define the product as follows:

  $$\begin{aligned} x^{\alpha }e_i\cdot x^{\beta }e_j= \left\{ \begin{array}{l} x^{\alpha +\beta }e_i\ \text {if}\ i=j\\ 0\ \text {otherwise}. \end{array}\right. \end{aligned}$$

  In the same way, we define the least common multiple as follows: if $c_1,c_2\in \mathbb {Z}$, then

  $$\begin{aligned} \text {lcm}(c_1x^{\alpha }e_i, c_1x^{\beta }e_j)= \left\{ \begin{array}{l} \text {lcm}(c_1x^{\alpha }, c_1x^{\beta })e_i=(\text {lcm}(c_1,c_2)\cdot \text {lcm}(x^{\alpha },x^{\beta }))e_i\ \text {if}\ i=j\\ 0\ \text {otherwise}. \end{array}\right. \end{aligned}$$

• 6. A monomial ordering on F is a total ordering < on $\text {Mon}(F)$ such that if $x^{\alpha }e_i$ and $x^{\beta }e_j$ are monomials in $\text {Mon}(F)$, and $x^{\gamma }$ any monomial in R, then

  $$x^{\alpha }e_i< x^{\beta }e_j\implies (x^{\alpha +\gamma })e_i < (x^{\beta +\gamma })e_j.$$

  Moreover, $x^{\alpha }e_i < x^{\beta }e_i$ if and only if $x^{\alpha }e_j < x^{\beta }e_j\ \forall \ i,j$.

• 7. Let < be a monomial ordering on F, let $f\in F\setminus \{0\}$, and let $f=cx^{\alpha }e_i+f^*$ be the unique decomposition of f with $c\in \mathbb {Z}\setminus \{0\}, x^{\alpha }e_i\in \text {Mon}(F)$, and $x^{\alpha }e_i>x^{\beta }e_j$ for any non-zero term $c^*x^{\beta }e_j$ occurring in $f^*$. We define the leading monomial, the leading coefficient, the leading term, and the tail of f as $Lm(f ):= x^{\alpha }e_i, Lc(f ):=c, Lt(f ):=cx^{\alpha }e_i, \text {tail}(f):=f-Lt(f)$ respectively.

• 8. For any sub-module $I\subset F$ (or ideal in the case $r=1$), we define $Lt(I):=\langle Lt(f)/f\in F\rangle \subset F$ the leading module (or leading ideal) of F.

• 9. An element $(h_1,\ldots ,h_s)\in R^s$ is called syzygy for $f_1,\ldots ,f_r\in R$ if $h_1f_1+\ldots ,h_sf_s = 0$.

Definition A.2

(S-vector). Let $f,g\in F=R^r=(\mathbb {Z}[x_1,\ldots ,x_n])^r$ be non-zero polynomial vectors and < be a monomial ordering. We define the S-vector of f and g in F as follows:

$$\begin{aligned} S(f,g)= \left\{ \begin{array}{ll} \displaystyle \frac{\text {Lc}(g)}{\text {Lc}(f)}m_{gf}\cdot f - m_{fg}\cdot g\ \text {if}\ \text {Lc}(f)\mid \text {Lc}(g)\\ m_{gf}\cdot f - \displaystyle \frac{\text {Lc}(f)}{\text {Lc}(g)}m_{fg}\cdot f\ \text {if}\ \text {Lc}(g)\mid \text {Lc}(f)\ \text {and}\ \text {Lc}(f)\not \mid \text {Lc}(g). \end{array}\right. \end{aligned}$$

Where $\text {m}_{gf}=\displaystyle \frac{\text {lcm}(\text {Lm}(f),\text {Lm}(g))}{\text {Lm}(f)}$ and $m_{fg}=\displaystyle \frac{\text {lcm}(\text {Lm}(f),\text {Lm}(g))}{Lm(g)}$.

If $F=R$ we call an S-vector by S-polynomial.

Special Gröbner bases

In this section we denote by $I=\langle f_1,\ldots ,f_s\rangle$ an $R-$submodule of $F=R^r=(\mathbb {Z}[x_1,\ldots ,x_n])^r$. We revisit the technique for computing special Gröbner bases for I introduced in [8,9,10]. Let $f,g\in F$ and < be a monomial ordering. If $Lc(f)=\gcd (\text {Lc}(f),\text {Lc}(g))\cdot a$ and $\text {Lc}(g)=\gcd (\text {Lc}(f),\text {Lc}(g))\cdot b$ with $\gcd (a,b)=1$, then by the fundamental theorem of arithmetic there exist prime numbers $p_1,\ldots ,p_r,q_1,\ldots ,q_s$ with $p_i\ne q_j\ \forall \ i,j$ and integers $l_1,\ldots ,l_r,m_1,\ldots ,m_s$ such that $a = p_1^{l_1}\cdots p_r^{l_r}$ and $b = q_1^{m_1}\cdots q_r^{m_r}$.

With these notations, we have the following:

Definition A.3

(Special S-vector). By a special S-vector of f and g we mean an S-vector S(f, g) defined over the localized ring $\mathbb {Z}_{[p_1\cdots p_r]}$ or $\mathbb {Z}_{[q_1\cdots q_s]}$ where $\mathbb {Z}_{[p_1\cdots p_r]}=\{\dfrac{c}{p_1^{n_1}\cdots p_r^{n_r}}\mid c\in \mathbb {Z}\ \text {and}\ n_1,\ldots ,n_r\in \mathbb {N}\}$

Remark A.4

The idea of the previous definition is described by the following tree

which means that if one of the leading coefficient divides the other, use directly the Definition A.2 with coefficients in $\mathbb {Z}$. If none of the leading coefficient divides the other, apply the Definition A.2 in any of $\mathbb {Z}_{[p_1\cdots p_r]}$ or $\mathbb {Z}_{[q_1\cdots q_s]}$ since in any of these localized ring, one of the leading coefficient divides the other.

Example A.5

Let $f=\begin{pmatrix} 12x^2y+y^2\\ 5xy^2+9 \end{pmatrix},\ g=\begin{pmatrix} 15x^3+7y^3\\ 4xy \end{pmatrix}\in R^2=(\mathbb {Z}[x,y])^2$. Let < be the degree reverse lexicographic ordering, then we have: $f=f_1e_1+f_2e_2,\ g=g_1e_1+g_2e_2$ with $f_1 = 12x^2y+y^2, f_2 = 5xy^2+9, g_1 = 15x^3+7y^3$ and $g_2 = 4xy$. Observe that $Lt(f_1) = 12x^2y > Lt(f_2) = 5xy^2$ then $Lt(f) = 12x^2ye_1$ and $Lt(g) = 15x^3e_1$. Since $\text {Lc}(f)=12,\ \text {Lc}(g)=15$ and $\gcd (\text {Lc}(f), \text {Lc}(g)) = 3$ then $\text {Lc}(f) = 3\cdot 4,\ \text {Lc}(g) = 3\cdot 5$. In this case

where $\mathbb {Z}_{[5]} = \{\dfrac{a}{5^{n_2}}\mid \ a\in \mathbb {Z}\ \text {and}\ n_2\in \mathbb {N}\}$ and $\mathbb {Z}_{[2]} = \{\dfrac{a}{2^{n}}\mid \ a\in \mathbb {Z}\ \text {and}\ n\in \mathbb {N}\}$.

• 1. Observe that 5 is a unit in $\mathbb {Z}_{[5]}$, in this ring 15 divides 12 and we have:

  $$S(f,g)=x\cdot f-\dfrac{4}{5}y\cdot g;$$

• 2. 2 is a unit in $\mathbb {Z}_{[2]}$ and in this ring 12 divides 15 and we have:

  $$S(f,g)=\dfrac{5}{4}x\cdot f-y\cdot g.$$

Example A.6

Let $I=\langle f_1=10xy+1, f_2=6x^2+3\rangle$ be an ideal of $\mathbb {Z}[x,y]$. We wish to compute a special Gröbner basis for I w.r.t the degree reverse lexicographic ordering $d_p$. Observe that $\text {Lc}(f_1)\not \mid \text {Lc}(f_2)$ and $\text {Lc}(f_2)\not \mid \text {Lc}(f_1)$, in this case we write: $\text {Lc}(f_1)=2\cdot 5$ and $\text {Lc}(f_2)=2\cdot 3$, we have:

Observe that in $\mathbb {Z}_{[5]},\ 5$ is invertible and $\text {Lc}(f_1)\mid \text {Lc}(f_2)$, we have in $R_1=\mathbb {Z}_{[5]}[x,y]$:

$$S(f_1,f_2)=\dfrac{3}{5}xf_1-yf_2=\dfrac{3}{5}x-3y.$$

Since $\dfrac{3}{5}x-3y$ is the remainder of the division of $S(f_1,f_2)$ by $\{f_1,f_2\}$, we set $f_3=\dfrac{3}{5}x-3y$. Observe that $Lc(f_1)$ and $Lc(f_3)$ are not compatible under the division in $R_1$. Since $Lc(f_1)=2\cdot 5$ and $Lc(f_3)=3\cdot \dfrac{1}{5}$, we can localize either by $2^{\mathbb {N}}$ or $3^{\mathbb {N}}$. Let us localize $R_1$ by $2^{\mathbb {N}}$, this gives $R_2=\mathbb {Z}_{[5.2]}[x,y]$. Note that in $R_2$ we have $Lc(f_1)\mid Lc(f_3)$ and:

$$S(f_1,f_3)=\dfrac{3}{50}f_1-yf_3=3y^2+\dfrac{3}{50}.$$

Since $3y^2+\dfrac{3}{50}$ is the remainder of $S(f_1,f_3)$ by $\{f_1,f_2,f_3\}$ then we set $f_4=3y^2+\dfrac{3}{50}$.

Observe that:

$$\begin{aligned} S(f_1,f_4)= & {} \dfrac{3}{10}yf_1-xf_4=-\dfrac{1}{10}f_3;\\ S(f_2,f_3)= & {} 10xf_3 - f_2=-3f_1;\\ S(f_2,f_4)= & {} y^2f_2 - 2x^2f_4=-\dfrac{1}{50}f_2+f_4\\ S(f_3,f_4)= & {} y^2f_3 - \dfrac{1}{5}xf_4=-\dfrac{1}{50}f_3-yf_4\\ \end{aligned}$$

Thus $G=\{f_1,f_2,f_3,f_4\}$ is a strong Gröbner basis for $S^{-1}I$ in $R_2$ where $S=2^{\mathbb {N}}\cdot 5^{\mathbb {N}}$, that is, G is a special Gröbner basis for I in R.

Example A.7

Let $N=\langle f_1,f_2\rangle$ be an $R-$submodule of $R^2=(\mathbb {Z}[x,y])^2$ with $f_1=\begin{pmatrix} 12x^2y+y^2\\ 5xy^2+9 \end{pmatrix},\ f_2=\begin{pmatrix} 15x^3+7y^3\\ 4xy \end{pmatrix}\in R^2$. Let < be the degree reverse lexicographic ordering and $<_M$ be a module ordering with priority given to monomials. We have:

$Lt(f) = 12x^2ye_1$ and $Lt(f_2) = 15x^3e_1$. Since $\text {Lc}(f_1)=12,\ \text {Lc}(f_2)=15$ and none of 12 and 15 divides the other in $\mathbb {Z}$ then consider: $\gcd (\text {Lc}(f_1), \text {Lc}(f_2)) = 3$ with $\text {Lc}(f_1) = 3\cdot 4$ and $\text {Lc}(f_2) = 3\cdot 5$. In this case we have

where $\mathbb {Z}_{[5]} = \{\dfrac{a}{5^{n_2}}\mid \ a\in \mathbb {Z}\ \text {and}\ n_2\in \mathbb {N}\}$ and $\mathbb {Z}_{[2]} = \{\dfrac{a}{2^{n}}\mid \ a\in \mathbb {Z}\ \text {and}\ n\in \mathbb {N}\}$.

• 1. Since 2 is a unit in $\mathbb {Z}_{[2]}$, then in this ring 12 divides 15 and we have:

  $$S(f_1,f_2)=\dfrac{5}{4}x\cdot f_1-y\cdot f_2 = \begin{pmatrix} -7y^4+\dfrac{5}{4}xy^2\\ \dfrac{25}{4}x^2y^2-4xy^2+\dfrac{5}{4}xy \end{pmatrix}.$$

  Since the remainder of the division’s algorithm is $S(f_1,f_2)$ then set

  $$f_3 = \begin{pmatrix} -7y^4+\dfrac{5}{4}xy^2\\ \dfrac{25}{4}x^2y^2-4xy^2+\dfrac{5}{4}xy \end{pmatrix}.$$

• 2. Observe that $Lt(f_3)=\dfrac{25}{4}x^2y^2e_3$ and $S(f_1,f_3) = S(f_2,f_3) = 0$. This means that $G=\{f_1,f_2,f_3\}$ is a strong Gröbner basis for $S^{-1}N$ as an $S^{-1}R-$submodule of $(S^{-1}\mathbb {Z}[x,y])^2$ with $S=\{2^0,2,2^2,2^3,\ldots ,\}$. In the other words, G is a special Gröbner basis for N in $R^2$.

Remark A.8

The Theorem 1.4 can be generalized for sub-modules of $R^s$. The following algorithm computes a weak Gröbner basis for $\mathbb {Z}(I)$.

Example A.9

Let $I=\langle f_1=10xy+1, f_2=6x^2+3\rangle$ be an ideal of $R=\mathbb {Z}[x,y]$. Let us compute a weak Gröbner basis for $\mathbb {Z}(I)$ w.r.t $d_p$. We have seen in the Example A.6 that $G=\{10xy+1,f_2 f_3=\dfrac{3}{5}x-3y, f_4=3y^2+\dfrac{3}{50}\}$ form a special Gröbner basis for I, using the Algorithm 3, the set $G'=\{f_1,f_2,5f_3,50f_4\}$ for a weak Gröbner basis for $\mathbb {Z}(I)$ in R w.r.t $d_p$.

Remark A.10

From Theorem 2.6 we give the following algorithm in the general case of modules. Observe that if $G=\{g_1,\ldots ,g_s\}$ is a special Gröbner basis for a sub-module $I\subset R^r$ with multiplicative subset S then each $g_i\in (S^{-1}R)^r$ that is, $g_i = (g_i^1,\ldots ,g_i^r)$ where $g_i^j\in S^{-1}R$ for $j=1,\ldots ,r$ and $i=1,\ldots ,s$.

Example A.11

In the Example 2.5 we have computed a strong Gröbner basis T for $\text {Syz}(f_1,f_3,f_4)$ in $S^{-1}\mathbb {Z}[x,y]$ where $S=2^{\mathbb {N}}\cdot 5^{\mathbb {N}}$. In the Example A.9 we have seen that $G'=\{g_1,g_3,g_4\}$ form a a minimal weak Gröbner basis for $\mathbb {Z}(I)$ in $R=\mathbb {Z}[x,y]$ with $g_1=f_1,g_3=5f_3$ and $g_4=50f_4$. Our gaol is to compute from each $G_{ji}$ obtained in the Example 2.5, the corresponding syzygies for $g_1,g_3,g_4$ using the Algorithm 4. We have:

• for $G_{13}$, we have $\dfrac{3}{50}f_1-yf_3-f_4=0$. Since $q_1=1, q_3=5, q_4=50$ are least common multiples of denominators of $f_1,f_3,f_4$ respectively and $b_1=50,b_3=b_4=1$ are least common multiples of denominators of components of $G_{13}$, then by the Algorithm 4 we have $q=\text {lcm}(q_1b_1,q_3b_3,q_4b_4)=50$ and $r_1=\dfrac{q}{q_1}=50, r_3=\dfrac{q}{q_3}=10$ and $r_4=\dfrac{q}{q_4}=1$. Thus $G_{13}'=(r_1h_1,r_3h_3,r_4h_4)=(3,-10y,-1)$ form a syzygy for $g_1,g_3,g_4$ in R since:

  $$3g_1-10yg_3-g_4=3(10xy+1)-10(3x-15y)-(3y^2+\dfrac{3}{50})=0.$$

• For each other $G_{ji}$ one can apply the same technique to obtain the corresponding syzygies in R.

With notations as in the Example A.11 we give the following example

Example A.12

We have seen in the Example 2.5 that $T=\{G_{13},G_{14},G_{34}\}$ form a a minimal strong Gröbner basis for $\text {syz}(f_1,f_3,f_4)$ in $(\mathbb {Z}_{2.5}[x,y])^3$. To obtain a weak Gröbner basis for $\text {syz}(g_1,g_3,g_4)$ observe that: $G_{13}=(\dfrac{3}{50},-y,-1)$ and denote by $a_1=50$ the least common multiple of all denominator occurring in $G_{13}$, $G_{14}=(\dfrac{3}{10}y,\dfrac{1}{10},-x)$ and denote $a_2=10$, and $G_{34}=(0,y^2+\dfrac{1}{50},y-\dfrac{1}{5}x)$ and denote $a_3=50$. Let $a=\text {lc }(a_1,a_2,a_3)=50$ and according to the Algorithm 5, the set $\{aG_{13},aG_{14}, aG_{34}\}=\{(3,-50y,-50), (15y,5,-50x), (0,50y^2+1,50y-10)\}$ form a weak Gröbner basis for $\mathbb {Z}(\text {Syz}(g_1,g_3,g_4))$ w.r.t the Schreyer’s ordering induced by $d_p$ and $f_1,f_3,f_4$.