Category Archives: Math

Polynomial division and Universal bases

Polynomial division

\(\DeclareMathOperator{\In}{In} \DeclareMathOperator{\ld}{Ld} \DeclareMathOperator{\kk}{\mathbb{K}} \DeclareMathOperator{\rnpos}{\mathbb{R}^n_{> 0}} \DeclareMathOperator{\rnzero}{\mathbb{R}^n_{\geq 0}} \DeclareMathOperator{\rr}{\mathbb{R}} \DeclareMathOperator{\scrB}{\mathcal{B}} \DeclareMathOperator{\scrI}{\mathcal{I}} \DeclareMathOperator{\scrJ}{\mathcal{J}} \DeclareMathOperator{\supp}{Supp} \DeclareMathOperator{\znpos}{\mathbb{Z}^n_{> 0}} \DeclareMathOperator{\znzero}{\mathbb{Z}^n_{\geq 0}} \DeclareMathOperator{\zz}{\mathbb{Z}}\)In this post we talk about division with respect to polynomials in more than one variables, which is a pretty cute algorithm that changed the face of a big part of mathematics, via e.g. Gröbner bases which we will also talk about. Dividing by polynomials in a single variable is straightforward – to divide \(f\) by \(g\), you pick a monomial \(m\) such that the highest degree terms of \(f\) and \(mg\) are equal, then repeat the same process with \(f – mg\) in place of \(f\). Since the degree, as a nonnegative integer, can not drop infinitely many times, this algorithm stops after a finitely many steps, and you end up with polynomials \(q\) (the quotient) and \(r\) (the remainder) such that \(f = qg + r\) and \(\deg(r) < \deg(f)\). To divide by polynomials in more than one variables, you need an analogue of the “highest degree monomial term” of the single-variable case. A natural resolution comes via monomial orders.

A monomial order is a binary relation \(\preceq\) on \(\znzero\) such that

  1. \(\preceq\) is a total order,
  2. \(\preceq\) is compatible with the addition on \(\znzero\), i.e. \(\alpha \preceq \beta\) implies \(\alpha + \gamma \preceq \beta + \gamma\) for all \(\gamma \in \znzero\), and
  3. \(0 \preceq \alpha\) for all \(\alpha \in \znzero\).

It turns out that in the presence of the first two conditions, the third one is equivalent to the following condition:

  • \(\preceq\) is a well order on \(\znzero\)

(For one direction: if \(\alpha \preceq 0\) for some \(\alpha \in \znzero \setminus \{0\}\), then \(\{\alpha, 2\alpha, 3\alpha, \ldots \}\) is an infinite set with no minimal element with respect to \(\preceq\). See [Mon21, Corollary B.47] for an elementary proof for the opposite direction.)

The support \(\supp(f)\) of a polynomial \(f \in \kk[x_1, \ldots, x_n]\), where \(\kk\) is a field, is the set of all \(\alpha \in \znzero\) such that \(x^\alpha\) appears in \(f\) with a nonzero coefficient. We write \(\delta_\preceq(f)\) for the maximal element of \(\supp(f)\) with respect to \(\preceq\). The leading term \(\ld_\preceq(f)\) of \(f\) with respect to \(\preceq\) is the monomial term \(cx^\alpha\) where \(\alpha = \delta_\preceq(f)\) and \(c\) is the coefficient of \(x^\alpha\) in \(f\). The division algorithm in \(\kk[x_1, \ldots, x_n]\) is the procedure for reducing a polynomial \(f\) by a finite ordered collection of polynomials \(g_1, \ldots, g_N\) as follows:

  1. If there is \(i\) such that \(\ld_\preceq(g_i)\) divides \(\ld_\preceq(f)\), then pick the smallest such \(i\), and replace \(f\) by \(f – hg_i\), where \(h := \ld_\preceq(f)/\ld_\preceq(g_i)\).
  2. Otherwise \(\ld_\preceq(f)\) is not divisible by \(\ld_\preceq(g_i)\) for any \(i\); in this case replace \(f\) by \(f – \ld_\preceq(f)\).
  3. Repeat.

Since \(\delta_\preceq(f)\) decreases at every step of the division algorithm, the well ordering property of \(\preceq\) implies that the algorithm stops after finitely many steps. It is clear that after the algorithm stops, one has \[f = \sum_i h_i g_i + r\] where either \(r = 0\), or for every \(i\), \(\ld_\preceq(g_i)\) does not divide any term of \(r\); we say that \(r\) is the remainder of \(f\) modulo division by the \(g_i\). This immediately implies the following result:

Theorem 1. Let \(g_1, \ldots, g_N\) be elements of an ideal \(I\) of \(\kk[x_1, \ldots, x_n]\) such that \(\ld_\preceq(g_1), \ldots, \ld_\preceq(g_N)\) generate the leading ideal \(\ld_\preceq(I)\) generated by \(\{\ld_\preceq(f): f \in I\}\). Then for all \(f \in \kk[x_1, \ldots, x_n],\) the following are equivalent:

  • \(f \in I\),
  • the remainder of \(f\) modulo \(g_1, \ldots, g_N\) is zero.

A Gröbner basis of \(I\) with respect to a monomial ordering \(\preceq\) is simply a finite set of elements \(g_1, \ldots, g_N \in I\) such that \(\ld_\preceq(g_i),\) \(1 \leq i \leq N,\) generate \(\ld_\preceq(I)\).

For later use we record a few corollaries of the division algorithm.

Corollary 1. Let \(I \supseteq J\) be ideals of \(\kk[x_1, \ldots, x_n]\). If \(I \supsetneqq J\), then \(\ld_\preceq(I) \supsetneqq \ld_\preceq(J)\) for every monomial order \(\preceq\).

Proof. If there are \(g_1, \ldots, g_N \in J\) such that \(\ld_\preceq(g_i)\), \(i = 1, \ldots, N\), generate \(\ld_\preceq(I)\), then Theorem 1 implies that \(g_1, \ldots, g_N\) generate \(I\), so that \(I = J\).

Corollary 2. Let \(I\) be an ideal of \(\kk[x_1, \ldots, x_n]\). There can not exist monomial orders \(\preceq_1, \preceq_2\) such that \(\ld_{\preceq_1}(I) \subsetneqq \ld_{\preceq_2}(I)\).

Proof. Write \(J_k := \ld_{\preceq_k}(I)\), \(k = 1, 2\). Assume to the contrary that \(J_1 \subsetneqq J_2\). Choose \(g_1, \ldots, g_N \in I\) such that \(\ld_{\preceq_2}(g_i)\), \(i = 1, \ldots, N\), generate \(J_1\). Choose \(f \in I\) such that \(\ld_{\preceq_2}(f) \not\in J_1\). Let \(r\) be the remainder of the division of \(f\) by \(g_1, \ldots, g_N\) with respect to \(\preceq_2\). Then \(r\) is a nonzero element of \(I\) such that \(\supp(r) \cap \supp(J_1) = \emptyset\). On the other hand, if \(h_1, \ldots, h_M \in I\) are such that \(\ld_{\preceq_1}(h_1), \ldots, \ld_{\preceq_1}(h_M)\) generate \(J_1\), then division of \(r\) by \(h_1, \ldots, h_M\) with respect to \(\preceq_1\) implies that \(r \not\in I\), which is a contradiction.

Starting from a Gröbner basis of an ideal, one can discard “redundant” elements (whose leading terms are divisible by the leading term of some other basis element) and then replace each remaining element by the remainder of the division by the others to obtain a reduced Gröbner basis, i.e. a Gröbner basis such that

  1. no monomial term of any basis element is divisible by the leading term of any other basis element, and
  2. the leading coefficient, i.e. the coefficient of the leading term, of each basis element is \(1.\)

Corollary 3. The reduced Gröbner basis is uniquely determined up to a permutation by the leading ideal. In particular,

  1. up to a permutation of the basis elements there is a unique reduced Gröbner basis for every monomial order, and
  2. if an ideal has the same leading ideal for two monomial orders, then it also has the same reduced Gröbner basis (up to permutations) for those monomial orders.

Proof. Indeed, fix two reduced Gröbner bases \(g_1, \ldots, g_N\) and \(h_1, \ldots, h_M\) of an ideal \(I\) with respect to a monomial order \(\preceq.\) For each \(h_j\), its leading term is divisible by the leading term of at least one \(g_i,\) but there can not be more than one such \(i,\) since then

  • either the leading term of both such \(g_i\) would be divisible by \(h_j,\)
  • or there would be \(j’ \neq j\) such that the leading term of \(h_{j’}\) divides the leading term of one of the \(g_i\), and therefore in turn divides the leading term of \(h_j\)

Each of these possibilities contradicts the reducedness of one of the above Gröbner bases. It follows that for each \(j \in \{1, \ldots, M\},\) there is a unique \(i \in \{1, \ldots, N\}\) such that \(\ld_\preceq(g_i)\) divides \(\ld_\preceq(h_j).\) Similarly for each $i \in \{1, \ldots, N\},$ there is a unique $j \in \{1, \ldots, M\}$ such that \(\ld_\preceq(h_j)\) divides \(\ld_\preceq(g_i).\) It follows that \(M = N\) and after reordering the \(g_i\) and the \(h_j\) if necessary, one can ensure that \(\ld_\preceq(g_i) = \ld_\preceq(h_i),\) \(i = 1, \ldots, N.\) Now fix \(i\) and divide \(h_i\) by \(g_1, \ldots, g_N.\) Since for \(j \neq i\) the leading term of \(g_j\) does not divide any monomial term of \(h_i,\) after the first step the remainder is \(h_i – g_i,\) and if \(h_i – g_i \neq 0,\) then at the second step the leading term of \(h_i – g_i\) must be divisible by the leading term of \(g_i.\) But this is impossible, since the exponent of each monomial term of \(h_i – g_i\) is smaller than \(\delta_\preceq(g_i) = \delta_\preceq(h_i).\) It follows that \(h_i = g_i,\) which proves the first assertion. For the second assertion, pick a monomial order \(\preceq’\) such that \(\ld_{\preceq’}(I) = \ld_\preceq(I).\) Let \(\alpha_i := \delta_{\preceq}(g_i),\) \(i = 1, \ldots, n.\) Fix \(i.\) Since \(\ld_{\preceq’}(I)\) is generated by \(x^{\alpha_1}, \ldots, x^{\alpha_N}\), it follows that \(\ld_{\preceq’}(g_i)\) is divisible by some \(x^{\alpha_j}.\) Since no monomial in the support of \(g_i\) is divisible by \(x^{\alpha_j}\) for \(j \neq i\) it must be divisible by \(x^{\alpha_i}.\) Since \(\alpha_i\) is also in the support of \(g_i,\) we must have that \(\delta_{\preceq’}(g_i) = \alpha_i.\) It follows that \(g_1, \ldots, g_N\) form a Gröbner basis of \(I\) with respect to \(\preceq’\) as well. It is clear that this is a reduced Gröbner basis, as required.

Universal bases for Leading ideals

In this section we prove that as \(\preceq\) varies over all possible monomial orders, there can be only finitely many leading ideals for a given ideal of a polynomial ring. The argument is elementary and very cute. I learned it from [Stu96, Chapter 1]. Mora and Robbiano [MoRobbiano88, Page 193] write that it is due to Alessandro Logar.

Theorem 2. Every ideal \(I\) of \(\kk[x_1, \ldots, x_n]\) has finitely many distinct leading ideals. In particular, there is a finite subset of \(I\) such that for every monomial order, the leading terms of polynomials in that subset generate the leading ideal of \(I.\)

Proof. Assume to the contrary that there is an infinite collection \(\scrJ_0\) of distinct leading ideals of \(I\). Pick an arbitrary nonzero element \(f_1 \in I\). Out of the finitely many monomial terms of \(f_1\), there must be one, say \(m_1\), such that \(\scrJ_1 := \{J \in \scrJ_0: \langle m_1 \rangle \subsetneqq J\}\) is infinite. Pick \(J_1 \in \scrJ_1\) and a monomial order \(\preceq_1\) such that to \(J_1 = \ld_{\preceq_1}(I)\). Since \(m_1 \in J_1\), there is \(g_1 \in I\) such that \(\ld_{\preceq_1}(g_1) = m_1\), and since \(\langle m_1 \rangle \neq J_1\), there is \(h_1 \in I\) such that \(\ld_{\preceq_1}(h_1) \not\in \langle m_1 \rangle.\) Let \(f_2\) be the remainder of \(h_1\) modulo division by \(g_1\). Then \(f_2\) is nonzero, and no monomial term in \(f_2\) belongs to \(\langle m_1 \rangle\). Since \(f_2\) has finitely many monomial terms, there must be a monomial term \(m_2\) of \(f_2\) such that \(\scrJ_2 := \{J \in \scrJ_1: \langle m_1, m_2 \rangle \subsetneqq J\}\) is infinite. Now choose \(J_2 \in \scrJ_2\) and a monomial order \(\preceq_2\) such that to \(J_2 = \ld_{\preceq_2}(I)\). As in the preceding case, there are \(g_{21}, g_{22}, h_2 \in I\) such that \(\ld_{\preceq_2}(g_{21}) = m_1\), \(\ld_{\preceq_2}(g_{22}) = m_2\), and \(\ld_{\preceq_2}(h_2) \not\in \langle m_1, m_2 \rangle.\) Define \(f_3\) to be the remainder of \(h_2\) modulo division by \(g_{21}, g_{22}\). Then \(f_3\) is nonzero, and no monomial term in \(f_3\) belongs to \(\langle m_1, m_2 \rangle,\) and we can proceed to define \(\scrJ_3\) as above. In this way we get an infinite strictly increasing chain of monomial ideals: \[ \langle m_1 \rangle \subsetneqq \langle m_1, m_2 \rangle \subsetneqq \langle m_1, m_2, m_3 \rangle \subsetneqq \cdots \] which violates the Noetherianity of \(\kk[x_1, \ldots, x_n]\) and completes the proof of the first assertion. The second assertion then follows from Corollary 3.

A universal (leading) basis for an ideal \(I\) of \(\kk[x_1, \ldots, x_n]\) is an ordered collection \(\scrB\) of elements in \(I\) such that for every monomial order \(\preceq\) on \(\znzero\), the leading ideal of \(I\) with respect to \(\preceq\) is generated by the leading terms of the elements in \(\scrB\). The above theorem implies that every ideal has a finite universal basis.

Every leading ideal comes from a positive weighted degree

Weighted degrees are natural generalizations of degree. Every \(\omega \in \rr^n\) defines a weighted degree, and for \(f \in \kk[x_1, \ldots, x_n]\) we write \(\ld_\omega(f)\) for the corresponding leading term of \(f\). Given an ideal \(I\) of \(\kk[x_1, \ldots, x_n]\) we similarly write \(\ld_\omega(I)\) for its leading ideal generated by \(\ld_\omega(f)\) for all \(f \in I\). In this section, also following [Stu96, Chapter 1], we prove the following result:

Theorem 3. For every ideal \(I\) of \(\kk[x_1, \ldots, x_n]\) and every monomial order \(\preceq\), there is \(\omega \in \znpos\) such that \(\ld_\omega(I) = \ld_\preceq(I)\) (in particular, this means that \(\ld_\omega(I)\) is a monomial ideal).

Remark. With a bit more work it can be more can be established (the proof is completely analogous to the proof of Theorem 3 from the next post): given a monomial order \(\preceq\) and polynomials \(g_1, \ldots, g_N ,\) there is \(\omega \in \znpos\) such that \[\ld_\omega(g_i) = \ld_\preceq(g_i),\ i = 1, \ldots, N.\]If in addition the \(g_i\) are elements of an ideal \(I\) such that \(\ld_\preceq(g_i),\) \(i = 1, \ldots, N,\) generate \(\ld_\preceq(I),\) then\[\ld_\omega(I) = \langle \ld_\omega(g_1), \ldots, \ld_\omega(g_N)\rangle = \langle \ld_\preceq(g_1), \ldots, \ld_\preceq(g_N)\rangle = \ld_\preceq(I)\]Moreover, every \(f \in I\) has an expression of the form \(f = \sum_i f_ig_i\) such that

  1. \(\delta_\preceq(f) = \max_\preceq\{\delta_\preceq(f_ig_i)\},\) and
  2. \(\omega(f) = \max\{\omega(f_ig_i)\}.\)

Proof of Theorem 3. Fix a Gröbner basis \(g_1, \ldots, g_N\) of \(I\) with respect to \(\preceq\). Write \[g_i = c_{i0} x^{\alpha_{i0}} + \cdots + c_{iN_i}x^{\alpha_{iN_i}}\]with \(\alpha_{i0} = \delta_\preceq(g_i)\). Define \[C := \{\omega \in \rnzero: \langle \omega, \alpha_{i0} – \alpha_{ij} \rangle \geq 0,\ i = 1, \ldots, N,\ j = 1, \ldots, N_i\}\]Note that \(C\) is a convex polyhedral cone. We claim that \(\dim(C) = n\). Indeed, \(C\) is dual to the cone \(C’\) generated by \(\alpha_{i0} – \alpha_{ij},\ i = 1, \ldots, N,\ j = 1, \ldots, N_i\), and the standard basis vectors \(e_1, \ldots, e_n\) of \(\rr^n\). If \(\dim(C) < n\), it would follow that \(C’\) is not strongly convex. Since \(C’\) is also rational, there would be nonnegative integers \(\lambda_{ij}\), not all zero, such that \[\sum_{i,j}\lambda_{ij}(\alpha_{i0} – \alpha_{ij}) = -\sum_k r_ke_k\]for some nonnegative integers \(r_1, \ldots, r_n\). Due to the second and third properties of a monomial order, it would follow that \[\sum_{i,j}\lambda_{ij} \alpha_{i0} \preceq \sum_{i,j}\lambda_{ij} \alpha_{ij}\]On the other hand, by construction \(\alpha_{i0} \succ \alpha_{ij}\) for each \(i,j\), so that \[\sum_{i,j}\lambda_{ij} \alpha_{i0} \succ \sum_{i,j}\lambda_{ij} \alpha_{ij}\]This contradiction shows that \(\dim(C) = n\), as claimed. In particular this implies (see e.g. [Mon21, Proposition V.15]) that the topological interior \(C^0\) of \(C\) has a nonempty intersection with \(\znpos\), and if \(\omega \in C^0 \cap \znpos\), then \(\ld_\omega(g_i) = c_{i0}x^{\alpha_{i0}}\). Since \(\ld_\preceq(I)\) is generated by \(c_{i0}x^{\alpha_{i0}}\), \(i = 1, \ldots, N,\) it follows that \(\ld_\omega(I) \supseteq \ld_\preceq(I)\). If this containment were proper, then Corollary 1 to Theorem 1 would imply that \[\ld_\preceq(\ld_\omega(I)) \supsetneqq \ld_\preceq(\ld_\preceq(I)) = \ld_\preceq(I)\]On the other hand, it is straightforward to check that \[\ld_\preceq(\ld_\omega(I)) = \ld_{\preceq_\omega}(I)\] where \(\preceq_\omega\) is the binary relation on \(\znzero\) defined as follows: \(\alpha \preceq_\omega \beta\) if and only if either \(\langle \omega, \alpha \rangle < \langle \omega, \beta \rangle\), or \(\langle \omega, \alpha \rangle = \langle \omega, \beta \rangle\) and \(\alpha \preceq \beta\). Since \(\omega \in \znzero\), it follows that \(\preceq_\omega\) is a monomial order, and taken toegther, the last two displays above contradict Corollary 2 to Theorem 1. Consequently, \(\ld_\omega(I) = \ld_\preceq(I)\), as required.

References

Lüroth’s theorem (a “constructive” proof)

Lüroth’s theorem (Lüroth 1876 for \(k = \mathbb{C}\), Steinitz 1910 in general). If \(k \subseteq K\) are fields such that \(k \subseteq K \subseteq k(x)\), where \(x\) is an indeterminate over \(k\), then \(K = k(g)\) for some rational function \(g\) of \(x\) over \(k\).

I am going to present a “constructive” proof of Lüroth’s theorem due to Netto (1895) that I learned from Schinzel’s Selected Topics on Polynomials (and give some applications to criteria for proper polynomial parametrizations). The proof uses the following result which I am not going to prove here:

Proposition (with the set up of Lüroth’s theorem). \(K\) is finitely generated over \(k\), i.e. there are finitely many rational functions \(g_1, \ldots, g_s \in k(x)\) such that \(K = k(g_1, \ldots, g_s)\).

The proof is constructive in the following sense: given \(g_1, \ldots, g_s\) as in the proposition, it gives an algorithm to determine \(g\) such that \(K = k(g)\). We use the following notation in the proof: given a rational function \(h \in k(x)\), if \(h = h_1/h_2\) with polynomials \(h_1, h_2 \in k[x]\) with \(\gcd(h_1, h_2) = 1\), then we define \(\deg_\max(h) := \max\{\deg(h_1), \deg(h_2)\}\).

Proof of Lüroth’s theorem

It suffices to consider the case that \(K \neq k\). Pick \(g_1, \ldots, g_s\) as in the proposition. Write \(g_i = F_i/G_i\), where

  • \(\gcd(F_i, G_i) = 1\) (Property 1).

Without loss of generality (i.e. discarding \(g_i \in k\) or replacing \(g_i\) by \(1/(g_i + a_i)\) for appropriate \(a_i \in k\) if necessary) we can also ensure that

  • \(\deg(F_i) > 0\) and \(\deg(F_i) > \deg(G_i)\) (Property 2).

Consider the polynomials \[H_i := F_i(t) – g_iG_i(t) \in K[t] \subset k(x)[t], i = 1, \ldots, s,\] where \(t\) is a new indeterminate. Let \(H\) be the greatest common divisor of \(H_1, \ldots, H_s\) in \(k(x)[t]\) which is also monic in \(t\). Since the Euclidean algorithm for computing \(\gcd\) respects the field of definition, it follows that:

  • \(H\) is also the greatest common divisor of \(H_1, \ldots, H_s\) in \(K[t]\), which means, if \(H = \sum_j h_j t^j\), then each \(h_j \in K\) (Property 3).

Let \(H^* \in k[x,t]\) be the polynomial obtained by “clearing the denominator” of \(H\); in other words, \(H = H^*/h(x)\) for some polynomial \(h \in k[x]\) and \(H^*\) is primitive as a polynomial in \(t\) (i.e. the greatest common divisor in \(k[x]\) of the coefficients in \(H^*\) of powers of \(t\) is 1). By Gauss’s lemma, \(H^*\) divides \(H^*_i := F_i(t)G_i(x) – F_i(x)G_i(t)\) in \(k[x,t]\), i.e. there is \(Q_i \in k[x,t]\) such that \(H^*_i = H^* Q_i \).

Claim 1. If \(\deg_t(H^*) < \deg_t(H^*_i)\), then \(\deg_x(Q_i) > 0\).

Proof of Claim 1. Assume \(\deg_t(H^*) < \deg_t(H^*_i)\). Then \(\deg_t(Q_i) > 1\). If in addition \(\deg_x(Q_i) = 0\), then we can write \(Q_i(t)\) for \(Q_i\). Let \(F_i(t) \equiv \tilde F_i(t) \mod Q_i(t)\) and \(G_i(t) \equiv \tilde G_i(t) \mod Q_i(t)\) with \(\deg(\tilde F_i) < \deg(Q_i)\) and \(\deg(\tilde G_i) < \deg(Q_i)\). Then \(\tilde F_i(t)G_i(x) – F_i(x) \tilde G_i(t) \equiv 0 \mod Q_i(t)\). Comparing degrees in \(t\), we have \(\tilde F_i(t)G_i(x) = F_i(x) \tilde G_i(t)\). It is straightforward to check that this contradicts Propeties1 and 2 above, and completes the proof of Claim 1.

Let \(m := \min\{\deg_\max(g_i): i = 1, \ldots, s\}\), and pick \(i\) such that \(\deg_\max(g_i) = m\). Property 2 above implies that \(\deg_t(H^*_i) = \deg_x(H^*_i) = m\). If \(\deg_t(H^*) < m\), then Claim 1 implies that \(\deg_x(H^*) < \deg_x(H^*_i) = m\). If the \(h_j\) are as in Property 3 above, it follows that \(\deg_\max(h_j) < m\) for each \(j\). Since \(H^* \not\in k[t]\) (e.g. since \(t-x\) divides each \(H_i\)), there must be at least one \(h_j \not \in k\). Since adding that \(h_j\) to the list of the \(g_i\) decreases the value of \(m\), it follows that the following algorithm must stop:

Algorithm

  • Step 1: Pick \(g_i := F_i/G_i\), \(i = 1, \ldots, s\), satisfying properties 1 and 2 above.
  • Step 2: Compute the monic (with respect to \(t\)) \(\gcd\) of \(F_i(t) – g_i G_i(t)\), \(i = 1, \ldots, s\), in \(k(x)[t]\); call it \(H\).
  • Step 3: Write \(H = \sum_j h_j(x) t^j\). Then each \(h_j \in k(g_1, \ldots, g_s)\). If \(\deg_t(H) < \min\{\deg_\max(g_i): i = 1, \ldots, s\}\), then adjoin all (or, at least one) of the \(h_j\) such that \(h_j \not\in k\) to the list of the \(g_i\) (possibly after an appropriate transformation to ensure Property 2), and repeat.

After the last step of the algorithm, \(H\) must be one of the \(H_i\), in other words, there is \(\nu\) such that \[\gcd(F_i(t) – g_i G_i(t): i = 1, \ldots, s) = F_{\nu}(t) – g_{\nu}G_{\nu}(t).\]

Claim 2. \(K = k(g_{\nu})\).

Proof of Claim 2 (and last step of the proof of Lüroth’s theorem). For a given \(i\), polynomial division in \(k(g_\nu)[t]\) gives \(P, Q \in k(g_\nu)[t]\) such that \[F_i(t) = (F_{\nu}(t) – g_{\nu}G_{\nu}(t))P + Q,\] where \(\deg_t(Q) < \deg_t(F_{\nu}(t) – g_{\nu}G_{\nu}(t))\). If \(Q = 0\), then \(F_i(t) = (F_{\nu}(t) – g_{\nu}G_{\nu}(t))P\), and clearing out the denominator (with respect to \(k[g_\nu]\)) of \(P\) gives an identity of the form \(F_i(t)p(g_\nu) = (F_{\nu}(t) – g_{\nu}G_{\nu}(t))P^* \in k[g_\nu, t]\) which is impossible, since \(F_{\nu}(t) – g_{\nu}G_{\nu}(t)\) does not factor in \(k[g_\nu, t]\). Therefore \(Q \neq 0\). Similarly, \[G_i(t) = (F_{\nu}(t) – g_{\nu}G_{\nu}(t))R + S,\] where \(R, S \in k(g_\nu)[t]\), \(S \neq 0\), and \(\deg_t(S) < \deg_t(F_{\nu}(t) – g_{\nu}G_{\nu}(t))\). It follows that \[F_i(t) – g_iG_i(t) = (F_{\nu}(t) – g_{\nu}G_{\nu}(t))(P – g_iR) + Q – g_iS.\] Since \(F_{\nu}(t) – g_{\nu}G_{\nu}(t)\) divides \(F_{i}(t) – g_{i}G_{i}(t)\) in \(k(x)[t]\) and since \(\deg_t(Q – g_iS) < \deg_t(F_{\nu}(t) – g_{\nu}G_{\nu}(t))\), it follows that \(Q = g_iS\). Taking the leading coefficients (with respect to \(t\)) \(q_0, s_0 \in k(g_\nu)\) of \(Q\) and \(S\) gives that \(g_i = q_0/s_0 \in k(g_\nu)\), as required to complete the proof.

Applications

The following question seems to be interesting (geometrically, it asks when a given polynomial parametrization of a rational affine plane curve is proper).

Question 1. Let \(k\) be a field and \(x\) be an indeterminate over \(k\) and \(g_1, g_2 \in k[x] \). When is \(k(g_1, g_2) = k(x)\)?

We now give a sufficient condition for the equality in Question 1. Note that the proof is elementary: it does not use Lüroth’s theorem, only follows the steps of the above proof in a special case.

Corollary 1. In the set up of Question 1, let \(d_i := \deg(g_i)\), \(i = 1, 2\). If the \(\gcd\) of \(x^{d_1} – 1, x^{d_2} – 1\) in \(k[x]\) is \(x – 1\), then $k(g_1, g_2) = k(t)$. In particular, if \(d_1, d_2\) are relatively prime and the characteristic of \(k\) is either zero or greater than both \(d_1, d_2\), then $k(g_1, g_2) = k(x)$.

Remark. Corollary 1 is true without the restriction on characteristics, i.e. the following holds: “if \(d_1, d_2\) are relatively prime, then $k(g_1, g_2) = k(x)$.” François Brunault (in a comment to one of my questions on MathOverflow) provided the following simple one line proof: \([k(x): k(g_1, g_2)]\) divides both \([k(x): k(g_i)] = d_i\), and therefore must be \(1\).

My original proof of Corollary 1. Following the algorithm from the above proof of Lüroth’s theorem, let \(H_i := g_i(t) – g_i(x)\), \(i = 1, 2\), and \(H \in k(x)[t]\) be the monic (with respect to \(t\)) greatest common divisor of \(H_1, H_2\).

Claim 1.1. \(H = t – x\).

Proof. It is clear that \(t-x\) divides \(H\) in \(k(x)[t]\), so that \(H(x,t) = (t-x)h_1(x,t)/h_2(x)\) for some \(h_1(x,t) \in k[x,t]\) and \(h_2(x) \in k[x]\). It follows that there is \(Q_i(x,t) \in k[x,t]\) and \(P_i(x) \in k[x]\) such that \(H_i(x,t)P_i(x)h_2(x) = (t-x)h_1(x,t)Q_i(x,t)\). Since \(h_2(x)\) and \((t-x)h_1(x,t)\) have no common factor, it follows that \(h_2(x)\) divides \(Q_i(x,t)\), and after cancelling \(h_2(x)\) from both sides, one can write \[H_i(x,t)P_i(x) = (t-x)h_1(x,t)Q’_i(x,t),\ i = 1, 2.\] Taking the leading form of both sides with respect to the usual degree on \(k[x,t]\), we have that \[(t^{d_i} – x^{d_i})x^{p_i} = a_i(t-x)\mathrm{ld}(h_1)\mathrm{ld}(Q’_i)\] where \(a_i \in k \setminus \{0\}\) and \(\mathrm{ld}(\cdot)\) is the leading form with respect to the usual degree on \(k[x,t]\). Since \(\gcd(x^{d_1} – 1, x^{d_2} – 1) = x – 1\), it follows that \(\mathrm{ld}(h_1)\) does not have any factor common with \(t^{d_i} – x^{d_i}\), and consequently, \(t^{d_i} – x^{d_i}\) divides \((t-x)\mathrm{ld}(Q’_i)\). In particular, \(\deg_t(Q’_i) = d_i – 1\). But then \(\deg_t(h_1) = 0\). Since \(H = (t-x)h_1(x)/h_2(x)\) is monic in \(t\), it follows that \(H = t – x\), which proves Claim 1.1.

Since both \(H_i\) are elements of \(k(g_1, g_2)[t]\), and since the Euclidean algorithm to compute \(\gcd\) of polynomials (in a single variable over a field) preserves the field of definition, it follows that \(H \in k(g_1, g_2)[t]\) as well (this is precisely the observation of Property 3 from the above proof of Lüroth’s theorem). Consequently \(x \in k(g_1, g_2)\), as required to prove Corollary 1.

References

  • Andrzej Schinzel, Selected Topics on Polynomials, The University of Michigan Press, 1982.

Publications

Overview

Starting from my PhD thesis Towards a Bezout-type Theory of Affine Varieties written at University of Toronto under the supervision of Pierre Milman, my research in math falls under two broad themes:

  • Compactification of affine varieties
  • Affine Bézout problem

I am in particular interested in one of the simplest cases of the first problem:

  • Compactification of $\mathbb{C}^2$

Book

How many zeroes?

Papers

Compactification of affine varieties

Compactification of $\mathbb{C}^2$

Preprints

  • Normal equivariant compactifications of $\mathbb{G}^2_a$ of Picard rank one, arXiv:1610.03563.
  • Mori dream surfaces associated with curves with one place at infinity, arXiv:1312.2168.
  • Analytic compactifications of $\mathbb{C}^2$ part II – one irreducible curve at infinity, arXiv:1307.5577.

Publications

Affine Bézout problem

Preprints

  • Intersection multiplicity, Milnor number and Bernstein’s theorem, arXiv:1607.04860

Publications (primarily announcements)

Simplest singularity on non-algebraic normal Moishezon surfaces

A classical question in complex analytic geometry is to understand when a given analytic space is algebraic (i.e. analytification of an algebraic scheme). A necessary condition for this to hold is that the transcendence degree of the field of global meromorphic functions must be equal to the dimension of the space, i.e. the space has to be Moishezon. For dimension 2, it is a classical result that it is also sufficient, provided the space is nonsingular (Chow and Kodaira, 1952).

In general it is not clear how to determine algebraicity of normal (singular) Moishezon surfaces and our understanding of non-algebraic Moishezon surfaces, more precisely what prevents them from being algebraic, remains incomplete (Schröer [Sch00] gives a necessary and sufficient for algebraicity, but it is not very suitable for computation in a given case). We [Mon16] gave an example of a non-algebraic normal Moishezon surface \(X\) which has the simplest possible singularity in the following sense: \(X\) has only one singular point \(P\), and the singularity at \(P\)

  1. has multiplicity 2 and geometric genus 1,
  2. is almost rational in the sense of [Ném07], and
  3. is a Gorenstein hypersurface singularity which is minimally elliptic (in the sense of [Lau77]).

The claim that singularity of \(X\) is the simplest possible is based on combining the preceding facts with the following observations:

  • a Moishezon surface whose singularities are rational (i.e. with geometric genus zero) is algebraic [Art62], and
  • minimally elliptic Gorenstein singularities form in a sense the simplest class of non-rational singularities.

The weighted dual graph of the resolution of singularity of \(X\) at \(P\) is of type \(D_{9,∗,0}\) in the classification of [Lau77] and the self-intersection number of its fundamental divisor is −2. It follows from [Lau77, Table 2] that the singularity at the origin of \(z^2 = x^5 + xy^5\) is also of the same type.

References

  • [Art62] Michael Artin, Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math., 84:485–496,
  • [Lau77] Henry B. Laufer, On minimally elliptic singularities, Amer. J. Math., 99(6):1257–1295, 1977.
  • [Mon16] Pinaki Mondal, Algebraicity of normal analytic compactifications of \(\mathbb{C}^2\) with one irreducible curve at infinity, Algebra & Number Theory, 10(8), 2016.
  • [Ném07] András Némethi, Graded roots and singularities, In Singularities in geometry and topology, pages 394–463. World Sci. Publ., Hackensack, NJ, 2007.
  • [Sch00] Stefan Schröer, On contractible curves on normal surfaces, J. Reine Angew. Math., 524:1–15, 2000.