The notion of the nonminimality degree of a stationary type was introduced by James Freitag and Rahim Moosa in the paper Bounding nonminimality and a conjecture of Borovik-Cherlin, J. Eur. Math. Soc. 27, 589 – 613 (2025). They were motivated by the problem of analyzing differential algebraic relations amongst solutions of differential equations and with Rémi Jaoui showed in When any three solutions are independent, Invent. math. (2022) 230: 1249 – 1265 that for an ordinary differential field $(k,\delta)$ of characteristic zero and an irreducible polynomial $P(x_0, x_1, \ldots, x_n) \in k[x_0, \ldots, x_n]$ with $n \geq 1$, if one can show that for any three distinct solutions $a_1$, $a_2$, and $a_3$ of the differential equation $P(x,\delta x, \ldots, \delta^n x) = 0$ that $a_1$, $a_2$, and $a_3$ are independent in the sense of forking independence or more concretely in the sense that the transcendence degree over $k$ of the field $k(a_1, \delta a_1, \ldots, \delta^{n-1} a_1, a_2, \delta a_2, \ldots, \delta^{n-1} a_2, a_3, \delta a_3, \ldots, \delta^{n-1} a_3)$ is $3n$, then all sets of solutions of that equation are independent. Moreover, they show that if the coefficients of $P$ are constant, then one may replace $3$ by $2$. In the follow up work James Freitag, Rémi Jaoui, Rahim Moosa, The degree of nonminimality is at most 2, Journal of Mathematical Logic, Vol. 23, No. 3 (2023) 2250031 (6 pages) the same three authors show that, in fact, relative to $\operatorname{DCF}_{0,m}$, the theory of differentially closed fields of characteristic zero with $m$ commuting derivations or $\operatorname{CCM}$, the theory of compact complex manifolds, if $p$ is a stationary type (over any set of parameters) with $U(p) < \omega$, then the nonminimality degree of $p$ is at most $2$. This theorem is used to improve a theorem of Matthew Delvilbiss and James Freitag that a generic differential equation of high enough degree relative to its order defines a strongly minimal set to the assertion that “high enough degree” means “degree at least six”.
Some of the key ideas derive from work of Rahim Moosa and Anand Pillay in Some model theory of fibrations and algebraic reductions, Selecta Math. New Ser. (2014) 20:1067 – 1082 where they gave a model theoretic account of some work on hyperkähler manifolds, isolating the notion of a type not admitting a proper fibration.
For at least the first part of this semester, we will be following the development of the theory of the nonminimality degree through these papers and related work. Let me set up some of the details here.
Minimal types are the building blocks of the finite Lascar rank part of a stable theory. Let us fix some stable theory $T$, a sufficiently large saturated model $\mathbb{U} \models T$, and $A \subsetneq \mathbb{U}$ a small set of parameters.
Definition: A type $p \in S(A)$ is algebraic if the set of realizations $p(\mathbb{U})$ is finite.
Definition: A type $p \in S(A)$ is minimal if for every small set $A \subseteq B \subsetneq \mathbb{U}$ there is a unique non-algebraic type $q \in S(B)$ with $p \subseteq q$.
Remark: Note that if $p$ is minimal then it is non-algebraic: take $B := A$, then the only type $q \in S(B)$ which extends $p$ is $p$ itself. Thus, by definition of minimality it must be that $p = q$ is not algebraic. Likewise, if $p$ is minimal, then it is stationary: consider a set $B$ with $A \subseteq B \subsetneq \mathbb{U}$. Let $q \in S(B)$ be the unique non-algebraic type extending $p$. If $r \in S(B)$ is any other extension of $p$, then $r$ is algebraic, and, hence, a forking extension of $p$ as $p$ is non-algebraic. By stability, there are non-forking extensions of $p$. Thus, $q$ must be a non-forking extension of $p$ and is the only one.
By definition, if $p \in S(A)$ is a non-minimal, non-algebraic stationary type then there is some set $B$ with $A \subseteq B \subsetneq \mathbb{U}$ and a forking extension $q \in S(B)$ of $p$ with $q$ non-algebraic. Let us consider a Morley sequence $(a_i)_{i=0}^\infty$ in $q$. As a general fact (See Lemma 2.28 in Pillay’s Geometric Stability Theory), the canonical base of $q$, $\operatorname{Cb}(q)$, is definable from the Morley sequence. Since $q$ forks over $A$, we have $a_0 \not \downarrow_A \operatorname{Cb}(q)$ which because $\operatorname{Cb}(q) \subseteq \operatorname{dcl}( \{ a_i : i \in \mathbb{Z}_+ \})$ gives $a_0 \not \downarrow_A \{ a_i : i \in \mathbb{Z}_+ \}$ and then by finite character of forking, we may find a minimal $n \in \mathbb{Z}_+$ so that $a_0 \not \downarrow_A a_1, \ldots, a_n$. This observation gives us the degree of nonminimality.
Definition: If $p \in S(A)$ is a non-minimal, non-algebraic stationary type then we define the nonminimality degree of $p$, $\operatorname{nmd}(p)$, to be the least $n$ for which there are $a_1, \ldots, a_n$ realizing $p$ for which $p$ has a non-algebraic, forking extension to $A a_1, \ldots, a_n$. By convention, we set $\operatorname{nmd}(p) = 0$ for $p$ a minimal or algebraic type.
Let us consider some examples.
Example: Let $p \in S(A)$ be a minimal type and let $k > 1$ be any natural number greater than $1$. Let $a_1, \ldots, a_{k+1}$ be a Morley sequence of length $2k-1$ in $p$ and set $q := p^{\otimes k} = \operatorname{tp}(a_1, \ldots, a_k/A)$. By the Lascar inequalities $U(q) = k > 1$. Note that $\operatorname{tp}(a_1, \ldots, a_k / a_2, \ldots, a_{k+1} A)$ is a non-algebraic forking extension of $q$ over a single realization of $q$. Thus, $\operatorname{nmd}(q) = 1$.
Let us introduce a defintion.
Definition: Let $p \in S(A)$ and $q \in S(A)$ be two complete types over the same parameter set $A$. We say that $p$ and $q$ are interalgebraic if we may find realizations $a$ and $b$ of $p$ and $q$, respectively, so that $\operatorname{acl}(Aa) = \operatorname{Ab}$.
Proposition: If $p$ and $q$ are interalgebraic stationary types, then $\operatorname{nmd}(p) = \operatorname{nmd}(q)$.
Proof: By the Lascar inequalities, $U(p) = U(q)$. Thus, if we are in the trivial case where $\operatorname{nmd}(p) =0$ (which is equivalent to $U(p) \leq 1$) then we also have $\operatorname{nmd}(q) = 0$. So we may assume that $U(p) > 1$ and $U(q) > 1$.
Let $a_0, a_1, \ldots, a_k$ be realizations of $p$ witnessing that $\operatorname{nmd}(p) \leq k$. That is, $a_o \not \downarrow_A a_1, \ldots, a_k$ and $a_0 \notin \operatorname{acl}(A a_1, \ldots, a_k)$. By interalgebraicity of $p$ and $q$ we may find $b_0, b_1, \ldots, b_k$ realizing $q$ so that $\operatorname{acl}(Aa_i) = \operatorname{acl}(Ab_i)$ for $0 \leq i \leq k$. Thus, $\operatorname{acl}(A b_1, \ldots, b_k) = \operatorname{acl}(A a_1, \ldots, a_k)$ cannot contain $b_0$, for if it did, because $a_0 \in \operatorname{acl}(A b_0)$ and likewise $b_0 \not \downarrow_A b_1, \ldots, b_k$. Hence, $\operatorname{nmd})(q) \leq \operatorname{nmd}(p)$. Arguing symmetrically with the roles of $p$ and $q$ reversed, we see that $\operatorname{nmd}(p) = \operatorname{nmd}(q)$. $\Box$
Examples with $\operatorname{nmd}(p) > 1$ are hard to come by. Here is one general procedure.
Construction: Let $G \times X \to X$ be a $2$-transitive group action definable in some totally transcendental theory. For example, the usual action of $\operatorname{PGL}_{n+1}(K)$ on $\mathbb{P}^n(K)$ where $K$ is algebraically closed would fit the bill. Let $p$ be the generic type of $X$ in the two sorted-structure $(G,X)$ having the group structure on $G$ and the action $G \times X \to X$. Then if $a_1$ and $a_2$ are distinct realizations of $p$ we have that $a_1 \downarrow a_2$ as the pair $(a_1,a_2)$ has the same type as a pair of indepenent realizations of $p$. If we take $X$ so that $U(p) > 1$ (e.g. take $n \geq 2$ in the standard example), then we see that $\operatorname{nmd}(p) \geq 2$.
The presence of a group action in the example with $\operatorname{nmd}(p) > 1$ is not an accident: nonminimality degree is studied through analyses, in the usual stability theoretic sense, and the nontrivial steps in these analyses involve binding groups.
Definition: A proper fibration of the stationary type $p = \operatorname{tp}(a/A) \in S(A)$ is given by some $A$-definable function $f$ on $p$ so that $b = f(a) \in \operatorname{dcl}(Aa) \smallsetminus \operatorname{acl}(A)$ with $a \notin \operatorname{acl}(Ab)$. Setting $q := f_*(p) = \operatorname{tp}(b/A)$ we have that $q$ is a nonalgebraic type and the fibres of $f$, such as $\operatorname{tp}(a/Ab)$, are also nonalgebraic.
Example: If $p \in S(A)$ and $q \in S(A)$ are non-algebraic stationary types (possibly equal to each other), then $p \otimes q$ has a proper fibration. Indeed, let $a \models p$ and $b \models q$ with $a \downarrow_A b$ so that $(a,b) \models p \otimes q$. Let $f:p \otimes q \to q$ be defined by $(x,y) \mapsto y$. In particular, $f((a,b)) = b$ and $f_*( (p \otimes q) ) = q$. We have that $(a,b) \notin \operatorname{acl}( A b)$ so that the fiber $\operatorname{tp}( (a,b) / A b)$ is non-algebraic. It follows from this example that if $p \in S(A)$ is any non-algebraic type and $k > 1$, then the Morley power $p^{\otimes k}$ has a proper fibration.
Proposition: If $p$ has a proper fibration, then $\operatorname{nmd}(p) = 1$.
Proof: Let $a$ realize $p$ and $b \in \operatorname{dcl}(Aa) \smallsetminus \operatorname{acl}(A)$ witness that $p$ has a proper fibration. Let $f:p \to q := \operatorname{tp}(b/A)$ be an $A$-definable function with $f(a) = b$. Let $a’$ be any realization of $\operatorname{tp}(a/Ab)$ which is not in $\operatorname{acl}(Aa)$. This is possible as $\operatorname{tp}(a/Ab)$ is not algebraic. Since $b \in \operatorname{dcl}(Aa) \smallsetminus \operatorname{acl}(A)$, we have $b \not \downarrow_A a$. As $b \in \operatorname{dcl}(Aa’)$, we have $a’ \not \downarrow_A a$. That is, $\operatorname{tp}(a’/Aa)$ is a forking extension of $\operatorname{tp}(a’/A) = p$. As $a’ \notin \operatorname{acl}(Aa)$, this type is nonalgebraic. That is, we have succeeded in finding a nonalgebraic forking extension of $p$ over a single realization of $p$; put another way: $\operatorname{nmd}(p) = 1$. $\Box$
Read contrapositively, a stationary type $p$ with $\operatorname{nmd}(p) > 1$ cannot have a proper fibration.
The main result of the Moosa-Pillay paper mentioned above descibes when a stationary type of finite Lascar rank has no proper fibration.
Theorem: If the $p$ is a stationary type with $0 < U(p) < \omega$ and $p$ does not have a proper fibration, then either there is a modular minimal type $q$ for which $p$ is interalgebraic with $q^{\otimes U(p)}$ or there is a non-locally modular type $r$ for which $p$ is almost $r$-internal.
We defer the proof of this theorem to a later lecture.
Combining the Moosa-Pillay theorem with the proposition relating proper fibrations to nonminimality degree, we obtain the following proposition.
Proposition: If $p \in S(A)$ is stationary with $U(p) < \omega$ and $\operatorname{nmd}(p) > 1$, then there is nonlocally modular minimal type $r$ so that $p$ is almost $r$-internal.
Proof: By the proposition on fibrations and nonminimality degree, $p$ has no proper fibration. By the Moosa-Pillay theorem, either there is a minimal type $r$ so that either (1) $r$ is modular and $p$ is interalgebraic with $r^{\otimes U(p)}$ or (2) $r$ is non-locally modular and $p$ is almost $r$-internal. We need to rule out the first case. Since $\operatorname{nmd}(p) > 1$, we must have $U(p) > 1$. By our observations on Morley products and fibrations, $r^{\otimes U(p)}$ has a proper fibration. Then by the proposition on proper fibrations and nonminimality degree, $\operatorname{nmd}(r^{\otimes U(p)}) = 1$. As $p$ and $r^{\otimes U(p)}$ are interalgebraic, it would follow that $\operatorname{nmd}(p) = \operatorname{nmd}(r^{\otimes U(p)}) = 1$, contrary to our hypothesis on $p$. Therefore, we must be in case (2). $\Box$
Remark: In later lectures, we will see how by replacing $p$ with a type $q$ which interalgebraic with $p$, we may upgrade from almost internality to a non-locally modular minimal type to outright internality.
Granting this, to study the case where $U(p) < \omega$ and $\operatorname{nmd}(p) > 1$, we may assume that $p$ is internal to some non-locally modular minimal type $r$.
Proposition: If $p \in S(A)$ is stationary, $r \in S(A)$ with
- $U(r) < U(p)$
- $r(\mathbb{U})$ eliminates imaginaries, and
- $p$ is internal to $r$,
then $\operatorname{nmd}(p) = \min \{ k \in \mathbb{Z}_+: p^{\otimes k} \not \perp^w r \}$.
Proof: Since $p$ is internal to $r$, for some $k$, we have that $p^{\otimes k} \not \perp^w r$. (This follows from the reflection principle.). Take $k$ minimal and let $(a_1, \ldots, a_k) \models p^{\otimes k}$ and $b \models r$ witnessing this non-weak orthogonality. That is, $a_1, \ldots, a_k \not \downarrow_A b$. Then $c := \operatorname{Cb}_A(a_1, \ldots, a_k/A b) \in \operatorname{dcl}(b) \smallsetminus \operatorname{acl}(A) \subseteq r(\mathbb{U})^{\operatorname{eq}} \smallsetminus \operatorname{acl}(A)$. Thus, by elimination of imaginaries for $r(\mathbb{U})$ there is some $d \in r(\mathbb{U}) \cap \operatorname{dcl}(A a_1, \ldots, a_k) \smallsetminus \operatorname{acl}(A)$.
Claim: $\operatorname{tp}(a_k/A a_1, \ldots, a_{k-1}) \to \operatorname{tp}(d/A a_1, \ldots, a_{k-1})$ is a proper fibration.
Proof of Claim: We already know that $\operatorname{tp}(d/A a_1, \ldots, a_{k-1})$ is non-algebraic by minimality of $k$ and a Lascar inequality computation gives that $\operatorname{tp}(a_k/A a_1, \ldots, a_{k-1} d)$ is not algebraic: $U(a_1, \ldots, a_{k-1}, c) \leq \sum_{j=1}^{k-1} U(a_j) + U(d) \leq (k-1) U(p) + U(r) < k U(p) = U(a_1, \ldots, a_k/A)$. Hence, $U(a_k/a_1, \ldots, a_{k-1} , c A) > 0$. $\spadesuit$
It follows from the proposition on fibrations and nonminimality degree that $\operatorname{nmd}(\operatorname{tp}(a_k/A a_1, \ldots, a_{k-1})) = 1$. That is, we can find some $a’ \equiv_{A a_1, \ldots, a_{k-1}} a_k$ so that $a_{k} \not \downarrow_{A a_1 \ldots, a_{k-1}} a’$ and $a_k \notin \operatorname{acl}(A a_1, \ldots, a_{k-1}, a’)$. It follows from transitivity that $a_k \not \downarrow_A a_1, \ldots, a_{k-1}, a’$. That is, $\operatorname{tp}(a_k/A a_1, \ldots, a_{k-1}, a’)$ is non-algebraic forking extension of $p$ over $A$ together with $k$ new realizations of $p$. Put another way, $\operatorname{nmd}(p) \leq k$. $\Box$
For a general totally transcendental theory, we might not have a good description of the nonlocally modular minimal types. However, for theories satisfying the Zilber trichotomy, we would know, essentially by definition, that each nonorthogonality class of nonlocally modular minimal types is represented by the generic type of a pure algebraically closed field. This holds, for example, for the theory $\operatorname{DCF}_{0,m}$ of differentially closed fields of characteristic zero with $m$ commuting derivations in which the every non-locally modular minimal type is nonorthogonal to the generic type of the field of absolute constants, that is, the field defined by $\delta_1(x) = \cdots = \delta_m(x) = 0$. It also holds for the multisorted theory $\operatorname{CCM}$ of compact complex manifolds in which we have sort $M$ for each compact complex manifold and basic predicats $R_Y$ on $X_1 \times \cdots \times X_m$ for each closed complex analytic set $Y \subseteq X_1 \times \cdots \times X_m$ where each $X_i$ is a complex manifold. In this theory, from the complex projective line $\mathbb{P}^1(\mathbb{C})$ we obtain the complex field definable as $\mathbb{P}^1(\mathbb{C}) \smallsetminus \{ \infty \}$ with the graphs of addition and multiplication being the restrictions of closed complex analytic subsets of $\mathbb{P}^1(\mathbb{C}) \times \mathbb{P}^1(\mathbb{C}) \times \mathbb{P}^1(\mathbb{C})$. Every nonlocally modular minimal type relative to $\operatorname{CCM}$ is non-orthogonal to the generic type of this copy of the field $(\mathbb{C},+,\cdot)$.
Thus, relative to these two theories, instances of nonminimality degree greater than one will come from internality to the generic type of an algebraically closed field of characteristic zero defined over the empty set and the specific value of the nonminimality degree can be read off from the difference between non-orthogonality and non-weak orthonality.
The following proposition, whose proof we will leave to a later lecture, expresses this connection. We will write $\mathfrak{g}$ for the generic type of the absolute constants or of the field of complex numbers depending on whether we are working $\operatorname{DCF}_{0,m}$ or $\operatorname{CCM}$. In the paper by Moosa and Freitag there is a weaker hypothesis on $\mathfrak{g}$ potentially allowing this proposition to be applied to other totally transcendental theories.
Proposition: If $p \in S(A)$ is stationary with $U(p) < \omega$ and $k := \operatorname{nmd}(p) > 1$, then there is a type $q \in S(A)$ with
- $p$ and $q$ are interalgebraic,
- $p$ is $\mathfrak{g}$-internal, and
- if $G := \operatorname{Aut}(q/r)^\circ$ is the connected component of the binding group $q$ over $r$ and $S := q$, then $(G,S)$ is a definable homogeneous space which is generically $(k-1)$-transitive, meaning that there is an orbit $O$ of $G$ on $S^{(k-1)}$ for which $\operatorname{RM}(S^{(k)} \smallsetminus O) < \operatorname{RM}(S^{(k)})$.
We then turn to studying such generically transitive actions.
Let us record the defintion.
Definition: Let $G$ be a group of finite Morley rank and $G \curvearrowright S$ be a definable action of $G$ on a definable set $S$. We say that the action is generically transitive if there is some $s \in S$ so that $\operatorname{RM}(S \smallsetminus Gs) < \operatorname{RM}(S)$. For each positive natural number $k$ there is an induced action $G \curvearrowright S^{(k)}$ of $G$ on the set of $k$-tuples of distinct elements of $S$ defined by $g \cdot (s_1, \ldots, s_k) := (g \cdot s_1, \ldots, g \cdots s_k)$. We say that $G \curvearrowright S$ is generically $k$-transitive if the action $G \curvearrowright S^{(k)}$ is generically transitive.
The conjecture of Borovik and Cherlin mentioned in the title of the Freitag-Moosa paper proposes a connection between $\dim(S)$ and the degree of generic transitivity, together with a characterization of the extremal case.
Conjecture: If $G \curvearrowright S$ is a generically $k$-transitive action of the group $G$ of finite Morley rank on the infinite definable set $S$, then $k \leq \operatorname{RM}(S) + 2$. Moreover, if $k = \operatorname{RM}(S) + 2$, then there is definable algebraically closed field $K$ and a definable isomorphism between this action and the usual action of $\operatorname{PGL}_{\operatorname{RM}(S) +1}(K)$ on $\mathbb{P}^{\operatorname{RM}(S)}(K)$.
While the Borovik-Cherlin conjecture is open in general, Freitag and Moosa prove that is true for group actions definable in $\operatorname{ACF}_0$. In a recent preprint, The Borovik-Cherlin conjecture holds in ACF, Ulla Karhumäki and Nick Ramsey announce a proof of the Borovik-Cherlin conjecture in algebraically closed fields with no restriction on the characteristic. These proofs will be discussed in later lectures.
With the Borovik-Cherlin conjecture established for group actions internal to $\mathfrak{g}$, Freitag and Moosa may establish that in general $\operatorname{nmd}(p) \leq U(p) +2$. Finer analyses in the paper with Jaoui bring the upper bound down to a constant.
