{"id":277,"date":"2025-09-03T21:14:12","date_gmt":"2025-09-03T21:14:12","guid":{"rendered":"https:\/\/wp.math.berkeley.edu\/model-theory\/?p=277"},"modified":"2025-09-09T21:26:40","modified_gmt":"2025-09-09T21:26:40","slug":"introduction-to-the-model-theory-of-difference-fields","status":"publish","type":"post","link":"https:\/\/wp.math.berkeley.edu\/model-theory\/2025\/09\/03\/introduction-to-the-model-theory-of-difference-fields\/","title":{"rendered":"Introduction to the model theory of difference fields"},"content":{"rendered":"<p>Before we get to the scientfic content, allow me to remind you that the location for the seminar has changed to <strong>891<\/strong> Evans and we meet Wednesdays, 3-4pm.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Definition<\/strong>:\u00a0 A difference field $(K,\\sigma)$ is a field given together with a distinguished field endomorphism $\\sigma:K \\to K$.<\/p>\n<p>The class of difference fields is naturally axiomatized by a first-order theory in the language $\\mathcal{L}(+,\\cdot,-,0,1,\\sigma)$ of rings augmented by a unary function symbol for the endormorphism.<\/p>\n<p><strong>Theorem:<\/strong> The theory of difference fields has a model companion called $\\operatorname{ACFA}$ for &#8220;Algebraically closed fields with a (generic) automorphism&#8221;.<\/p>\n<p>Before we discuss the proof and attributions of this theorem, let us note how this theorem suggests a more general problem on theories of expansions by endomorphisms.<\/p>\n<p>Let $\\mathcal{L}$ be some first-order language and let $T$ be a theory which is assumed to be the model companion of it universal consequences $T_\\forall$.\u00a0 The example to keep in mind is $T = \\operatorname{ACF}$.\u00a0 \u00a0 In the language $\\mathcal{L}(\\sigma)$ expanding $\\mathcal{L}$ by a unary function symbol $\\sigma$ let $T_\\sigma$ be the theory $T$ together with axioms expressing that $\\sigma$ is an $\\mathcal{L}$-automorphism.\u00a0 \u00a0 \u00a0We say that &#8220;$T_A$ exists&#8221; if $T_\\sigma$ has a model companion. On general grounds, if $T_A$ exists, then it is the theory of the class of existentially closed models of $T_\\sigma$.\u00a0 \u00a0 \u00a0The theory on $\\operatorname{ACFA}$ says that $\\operatorname{ACF}_A$ exists.\u00a0 \u00a0However, there are many cases in which it is known that $T_A$ does not exist.<\/p>\n<ul>\n<li>\n<div><span class=\"font-weight-bold right-space\">Kikyo, Hirotaka, <a href=\"https:\/\/www.cambridge.org\/core\/journals\/journal-of-symbolic-logic\/article\/model-companions-of-theories-with-an-automorphism\/A56D072CB1F8B600E3DA2C2C232EFC8B\">Model companions of theories with an automorphism<\/a>, JSL 65 (2000), no. 3, 1215 &#8211; 1222 shows that if $T$ is unstable and NIP or stable with the finite cover property, or if $T$ is unstable and $T_\\sigma$ has the amalgamation property, then $T_A$ does not exist.<\/span><\/div>\n<\/li>\n<li>\n<div>\n<div><span data-testid=\"translated-paging-punctuation\"> Hirotaka Kikyo and Saharon Shelah, <a href=\"https:\/\/doi.org\/10.2178\/jsl\/1190150038\">The strict order property and generic automorphisms<\/a>, JSL\u00a0<strong>67<\/strong> (2002), no. 1, 214 &#8211; 216 show that $T_A$ does not exist when $T$ has the strict order property.<\/span><\/div>\n<\/div>\n<\/li>\n<li>\n<div class=\"d-inline-block mt-0 bibliography\">\n<div>\n<div><span data-testid=\"translated-paging-punctuation\">John Baldwin and Saharon Shelah, <a href=\"http:\/\/DOI: 10.1305\/ndjfl\/1063372196\">Model companions of $T_{\\operatorname{Aut}}$ for stable $T$,<\/a> Notre Dame J. Formal Logic\u00a0<strong>42\u00a0<\/strong>(2001), no. 3, 129 &#8211; 142 determine for which stable theories $T_A$ exists.<\/span><\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/li>\n<\/ul>\n<p>In the positive direction, Zo\u00e9 Chatzidakis and Anand Pillay, <a href=\"https:\/\/doi.org\/10.1016\/S0168-0072(98)00021-9\">Generic structures and simple theories<\/a>, APAL <strong>95<\/strong> (1998), pp. 71 &#8211; 92 show that $T_A$ exists, for instance, when $T$ is strongly minimal and has the definable multiplicity property.<\/p>\n<p>The exact dividing line for when $T_A$ exists remains unknown and there are many cases in which something like $T_A$ may be shown to exist when $T_\\sigma$ is strengthened to require further universal conditions on $\\sigma$. For example, if $T$ is the theory of nontrivially valued algebraically closed fields of residue characteristic zero and we define $T&#8217;_\\sigma$ to be $T_\\sigma$ together with the axiom that $(\\forall x) v(\\sigma(x)) = v(x)$, then $T&#8217;_\\sigma$ has a model companion.<\/p>\n<p>Harper Wells has offered to present details about $T_A$ later this term.<\/p>\n<p>The name &#8220;algebraically closed fields with a generic automorphism&#8221; suggests that in every model of ACFA, the endomorphism $\\sigma$ is an automorphism and that the underlying field is algebraically closed.\u00a0 Both of these asssertions are true, though these two conditions do not suffice to axiomatize ACFA.<\/p>\n<p><strong>Proposition:<\/strong>\u00a0 If $(K,\\sigma)$ is an existentially closed difference field, then $K$ is algebraically closed.<\/p>\n<p><strong>Proof:\u00a0<\/strong> Let $(K,\\sigma)$ be an existentially closed difference field.\u00a0 Let $L := K^\\text{alg}$ be an algebraic closure of $K$.\u00a0 Then there is an extension $\\tau:L \\to L$ of $\\sigma$ to $L$.\u00a0 To show this it suffices to see (by topological compactness of $\\text{Hom}(L,L)$ or the compactness theorem of first-order logic) that for each finite subextension $K \\leq M \\leq L$ that there is an extension $\\tau:M \\to L$ of $\\sigma$.\u00a0 \u00a0Working recursively, it suffices to consider the case that $M = K[a]$ is a simple extension.\u00a0 Let $P(x) \\in K[x]$ be the minimal monic polynomial of $a$ over $K$.\u00a0 Then $P^\\sigma(x) \\in K[x]$ is also a monic polynomial and thus has a zero $b$ in $L$. Using the universal mapping property of the polynomial ring, there is a unique homomorphism $\\rho:K[x] \\to L$ which restricts to $\\sigma$ on $K$ and takes $x$ to $b$.\u00a0 \u00a0The image of $P(x)$ under $\\rho$ is $P^\\sigma(b) = 0$.\u00a0 Hence, $\\rho$ induces a map $\\tau:L \\cong K[x]\/(P) \\to L$.\u00a0 \u00a0 Thus, $(K,\\sigma)$ extends to an algebraically closed difference field $(K,\\tau)$.\u00a0 For any nonconstant polynomial $Q \\in K[x]$ there is some $a \\in L$ with $Q(a) = 0$.\u00a0 That is, $(L,\\tau) \\models (\\exists x) Q(x) = 0$.\u00a0 As $(K,\\sigma)$ is existentially closed, $(K,\\sigma) \\models\u00a0 (\\exists x) Q(x) = 0$.\u00a0 That is, $K$ is algebraically closed. $\\Box$<\/p>\n<p><strong>Proposition:\u00a0<\/strong> If $(K,\\sigma)$ is an existentially closed difference field, then $\\sigma$ is an automorphism.<\/p>\n<p><strong>Proof:\u00a0<\/strong> \u00a0 The map $\\iota:(K,\\sigma) \\to (L,\\tau) := (K,\\sigma)$ given $x \\mapsto \\sigma(x)$ is an embedding of difference fields.\u00a0 \u00a0For each $a \\in K$, we have that $(L,\\tau) \\models (\\exists x) \\sigma(x) = a$ as in $L$, $a$ is interpreted as $\\sigma(a)$ and $\\tau(a) = \\sigma(a) = \\iota(a)$.\u00a0 Thus, as $(K,\\sigma)$ is existentially closed, $(K,\\sigma) \\models (\\exists x) \\sigma(x) =a$.\u00a0 That is, $\\sigma(K) = K$ so that on $K$, $\\sigma$ is an automorphism.\u00a0 $\\Box$<\/p>\n<p>Early in the development of the model theory of difference fields, it was observed (by Lou van den Dries, I think, though let me confirm!) the fixed field $F$ of $\\sigma$ in an existentially closed difference field $(K,\\sigma)$ is pseudofinite.<\/p>\n<p><strong>Definition:\u00a0<\/strong> A field $F$ is\u00a0<em>pseudofinite\u00a0<\/em>if it is an infinite model of $\\text{Th}(\\{ K~:~K \\text{ a finite field } \\})$, the theory of all finite fields.\u00a0 Equivalently, $F$ is pseudofinite if there is a nonprincipal ultrafilter $U$ on the set of prime powers for which $F \\equiv \\prod_U \\mathbb{F}_q$.<\/p>\n<p>James Ax produces a first-order axiomatization of the theory of pseudofinite fields in <a href=\"https:\/\/doi.org\/10.2307\/1970573\">The Elementary Theory of Finite Fields<\/a>, Annals of Mathematics, <strong>88<\/strong>, no. 2, September 1968, pp. 239 &#8211; 271.<\/p>\n<p><strong>Theorem<\/strong> (Ax):\u00a0 A field $F$ is pseudofinite if and only if<\/p>\n<ul>\n<li>$F$ is perfect,<\/li>\n<li>$\\text{Gal}(F^\\text{sep}\/F) \\cong \\widehat{\\mathbb{Z}}$, and<\/li>\n<li>$F$ is PAC: pseudo-algebraically closed<\/li>\n<\/ul>\n<p>The first of these conditions is in the usual sense of field theory:\u00a0 either $F$ has characteristic zero or it has characteristic $p &gt; 0$ and $F^p = F$.\u00a0 \u00a0In the presence of perfection, the second condition may be re-expressed by saying that for each positive integer $n \\in \\mathbb{Z}_+$ there is a unique extension of fields $F_n\/F$ with $[F_n:F] = n$.\u00a0 Each of these conditions is naturally expressed by a first-order theory.\u00a0 For example, perfection is given by the countably many sentences $$\\overbrace{1 + 1 + \\cdots + 1}^{p \\text{ times }} = 0 \\to (\\forall x)(\\exists y) y^p = x$$ We may describe field extensions of a fixed degree $n$ by quantifying over the coefficients of monic polynomials of degree $n$.\u00a0 For example, to say that there is an extension of degree $n$ we assert that there is an irreducible monic polynomial of degree $n$ which could be expressed as $$(\\exists a_0) \\cdots (\\exists a_{n-1}) \\left(\u00a0 \\bigwedge_{1 \\leq m &lt; n} (\\forall b_0) \\cdots (\\forall b_m) (\\forall c_0) \\cdots\u00a0 (\\forall c_{n-m-1}) \\bigvee_{\\ell=0}^{n-1}\u00a0 \\sum_{i+j=\\ell} b_i c_j \\neq a_\\ell\u00a0 \u00a0 \u00a0 \\right)$$\u00a0 Alternatively, one could more directly work with the family of $F$-algebras of dimension $n$ by describing these with $n \\times n \\times n$ arrays from $F$ giving the multiplication and then relativize the field axioms to these interpreted structures to isolate the fields.\u00a0 Maps between these field extensions are described by linear maps, that is, by matrices.\u00a0 So, one could express that all degree $n$ extensions are isomorphic.<\/p>\n<p>It is less clear how to express the PAC condition in first-order logic.\u00a0 In the definition of PAC we were a little sloppy about the meaning of &#8220;variety&#8221;.\u00a0 Nothing would be lost by taking this term to mean &#8220;embedded affine variety&#8221;.\u00a0 \u00a0Such varieties are described by systems of polynomial equations, which may be taken to be finite systems of equations by Noetherianity of the polynomial rings.\u00a0 For the natural statement of the PAC condition, one should show that given variables $\\mathbf{x} = (x_1, \\ldots, x_n)$ and $\\mathbf{y} = (y_1, \\ldots, y_m)$ and polynomials $F_1(\\mathbf{x},\\mathbf{y}), \\ldots, F_\\ell(\\mathbf{x},\\mathbf{y}) \\in \\mathbb{Z}[\\mathbf{x},\\mathbf{y}]$ there is a formula $\\theta(\\mathbf{y}) = \\theta_{\\mathbf{F}}$ so that for any field $F$ and tuple $\\mathbf{b} \\in F^{\\mathbf{y}}$ we have $F \\models \\theta(\\mathbf{b})$ if and only if the ideal generated by $F_1(\\mathbf{x},\\mathbf{b}), \\ldots, F_\\ell(\\mathbf{x},\\mathbf{b})$ in $F^\\text{alg}[\\mathbf{x}]$ is prime.\u00a0 \u00a0If one proves this assertion in the case that $F = F^\\text{alg}$, then using quantifier elimination for algebraically closed fields, one sees that $\\theta$ may be taken to be quantifier-free and then it will continue to have the requisite properties for arbitrary fields.\u00a0 \u00a0That such formulas exist was shown by Lou van den Dries and Karsten Schmidt in <a href=\"https:\/\/link.springer.com\/article\/10.1007\/BF01388493\">Bounds in the theory of polynomial rings over fields: A nonstandard approach<\/a>, Inventiones Math.\u00a0<strong>76\u00a0<\/strong>(1984), no. 1, 77 &#8211; 91.\u00a0 \u00a0The theorem is proven as a corollary of general result that if $K$ is any field and $U$ is an ultrafilter on some set $I$, then the algebra $(K[x_1,\\ldots,x_n])^U$ is a faithfully flat extension of $K^U[x_1, \\ldots, x_n]$.\u00a0 An alternative more elementary proof was given by Will Johnson in the appendix to <a href=\"https:\/\/doi.org\/10.1112\/jlms.12002\">Differential Chow varieties exist<\/a>, J. LMS (2),\u00a0<strong>95<\/strong> (2017), no. 1, 128 &#8211; 156.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Theorem:\u00a0<\/strong> If $(K,\\sigma)$ is an existentially closed difference field and $F := \\operatorname{Fix}(\\sigma,K) := \\{ x \\in K : \\sigma(x) = x \\}$, then $F$ is pseudofinite.<\/p>\n<p><strong>Proof:\u00a0<\/strong>First, we check that $F$ is perfect. If the characteristic of $K$ is zero, there is nothing to check.\u00a0 Suppose that the characteristic of $K$ is $p &gt; 0$.\u00a0 Let $a \\in F$.\u00a0 Since $K$ is algebraically closed, there is some $b \\in K$ with $b^p = a$.\u00a0 We compute that $\\sigma(b)^p = \\sigma(b^p) = \\sigma(a) = a = b^p$.\u00a0 Since the map $x \\mapsto x^p$ is injective on fields of characteristic $p$, we conclude that $\\sigma(b) = b$.\u00a0 That is, $b \\in F$.\u00a0 As $a \\in F$ was arbitrary, we see that $F = F^p$.\u00a0 That is, $F$ is perfect.<\/p>\n<p>For the second condition, let $n \\in \\mathbb{Z}_+$ and set $R := K[x_0,\\ldots,x_{n-1}]$.\u00a0 Let $L := \\mathcal{Q}(R) = K(x_0,\\ldots,x_{n-1})$ be the field of fractions of $R$.\u00a0 From the universal mapping property of the polynomial ring, there is a unique map $\\tau:R \\to L$ for which $\\tau \\upharpoonright K = \\sigma$ and $\\tau(x_i) = x_{i+1 \\mod n }$ for $0 \\leq i &lt; n$.\u00a0 \u00a0More concretely, if $P(x_0, \\ldots, x_{n-1}) \\in R$, then $\\tau(P) = P^\\sigma (x_1, x_2, \\ldots, x_{n-1}, x_0)$.\u00a0 \u00a0In particular, if $P \\neq 0$, then $\\tau(P) \\in L^\\times$.\u00a0 Thus, from the universal property of the localization, $\\tau$ extends uniquely to a map $\\tau:L \\to L$.\u00a0 That is, $(L,\\tau)$ is a difference field extension of $(K,\\sigma)$.<\/p>\n<p>The element $x_0 \\in L$ satisfies $\\tau^n(x_0) = x_0$.\u00a0 Let $P(X) := \\prod_{i=0}^{n-1} (X &#8211; x_i)$.\u00a0 Then $P \\in \\operatorname{Fix}(\\tau,L)[X]$ and its roots, $\\{ x_i : 0 \\leq i &lt; n \\}$,\u00a0 form one orbit under the action of the subgroup of $\\operatorname{Gal}(\\operatorname{Fix}(\\tau,L)^\\text{alg}\/\\operatorname{Fix}(\\tau,L))$.\u00a0 Hence, $[\\operatorname{Fix}(\\tau,L)(x_0):\\operatorname{Fix}(\\tau,L)] = n$.\u00a0 \u00a0That is, $$(L,\\tau) \\models (\\exists x) [\\operatorname{Fix}(\\sigma)(x):\\operatorname{Fix}(\\sigma)] = n$$ which implies that the same is true of $(K,\\sigma)$ as it is existentially closed in $(L,\\tau)$.\u00a0 \u00a0On general grounds, if $a \\in K$ and $[F(a):F] = n$, then $\\sigma^n(a) = a$.\u00a0 Indeed, if $P(X) \\in F[X]$ is the minimal monic polynomial of $a$, then $0 = \\sigma(0) = \\sigma(P(a))= P^\\sigma(\\sigma(a)) = P(\\sigma(a))$ where the last equality holds as $P$ is defined over $F$.\u00a0 \u00a0Thus, the set $\\{ \\sigma^i(a) : i \\in \\mathbb{Z} \\}$ has size exactly $n$, which implies in particular that $a = \\sigma^n(a)$.\u00a0 Therefore, $\\operatorname{Fix}(\\sigma^n)$ is the unique extension of $F$ of degree $n$.<\/p>\n<p>Finally, let us check that $F$ is PAC.\u00a0 Let $X \\subseteq \\mathbb{A}_F^n$ be an absolutely irreducible embedded affine variety over $F$.\u00a0 Let $g_1, \\ldots, g_m$ be generators of $I(X) \\subseteq \\mathbb{A}_F^n$.\u00a0 Since $X$ is absolutely irreducible, $(g_1, \\ldots, g_m) \\subseteq K[x_1, \\ldots, x_n]$ remains prime.\u00a0 Let $L := \\mathcal{Q}(K[x_1, \\ldots, x_n]\/(g_1, \\ldots, g_m))$ be the field of the ring $K[x_1, \\ldots, x_n]\/(g_1, \\ldots, g_m)$.\u00a0 Define $\\tau:K[x_1, \\dots, x_n] \\to L$ using the universal mapping property of polynomial rings via $\\tau \\upharpoonright K := \\sigma$ and $\\tau(x_i) = \\frac{x_i + (g_1, \\ldots, g_m)}{1}$ for $1 \\leq i \\leq n$.\u00a0 \u00a0The kernel of $\\tau$ is $(g_1, \\ldots, g_m)$ (NOTE: we are using the fact that $X$ is defined over $F$: for $h = \\sum h_\\alpha x^\\alpha \\in K[x_1, \\ldots, x_n]$, the image of $h$ under $\\tau$ is $h^\\sigma + (g_1, \\ldots, g_m) = \\sum \\sigma(h_\\alpha) x^\\alpha + (g_1, \\ldots, g_m)$.\u00a0 Thus, $\\tau(h) = 0$ if and only if $h^\\sigma \\in (g_1, \\ldots, g_m)$, which, because $\\sigma$ is an automorphism of $K$, is equivalent to $h \\in (g_1^{\\sigma^{-1}}, \\ldots, g_m^{\\sigma^{-1}}) = (g_1, \\ldots, g_m)$.).\u00a0 Thus, $\\tau$ extends to a map of fields $\\tau:L \\to L$.\u00a0 That is, $(K,\\sigma) \\subseteq (L,\\tau)$ is an extension of difference fields. $(L,\\sigma) \\models (\\exists x)\u00a0 x \\in X(\\operatorname{Fix}(\\sigma))$.\u00a0 Hence, as $(K,\\sigma)$ is existentially closed, it also satisfies this conclusion.\u00a0 $\\Box$<\/p>\n<p>&nbsp;<\/p>\n<p>This proposition suggested an ambitious conjecture which was subsequently proven to be true.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Theorem:\u00a0<\/strong> If $U$ is a nonprincipal ultrafilter on the set of prime power, then $\\prod_U (\\mathbb{F}_q^\\text{alg},x \\mapsto x^q)$ is an existentially closed difference field.\u00a0 In fact, if $(K,\\sigma)$ is an existentially closed difference field, then there is a nonprincipal ultrafilter $U$ on the set of prime powers for which $\\prod_U (\\mathbb{F}_q^\\text{alg},x \\mapsto x^q) \\equiv (K,\\sigma)$.<\/p>\n<p>This theorem was announced by Macintyre in the late 1990s, though his proof has not been made available.\u00a0 An arXiv posting of Hrushovski, <a href=\"https:\/\/arxiv.org\/abs\/math\/0406514\">The Elementary Theory of the Frobenius<\/a>, is devoted to its proof through a twisted version of Deligne&#8217;s theorem on rational points on varieties in finite fields proven with a mix of difference geometry, the model theory of valued fields, and \u00e9tale cohomology, amongst other methods.\u00a0 \u00a0Shuddhodan and Varshavsky, The Hrushovski-Lang-Weil estimates, Alg. Geom. <strong>9<\/strong> (2022), no. 6. 651 &#8211; 687, prove the estimates with arguments closer to Deligne&#8217;s method.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div data-v-43e39059=\"\">\n<div data-v-25fb2619=\"\" data-v-43e39059=\"\"><\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Before we get to the scientfic content, allow me to remind you that the location for the seminar has changed to 891 Evans and we meet Wednesdays, 3-4pm. &nbsp; Definition:\u00a0 A difference field $(K,\\sigma)$ is a field given together with &hellip; <a href=\"https:\/\/wp.math.berkeley.edu\/model-theory\/2025\/09\/03\/introduction-to-the-model-theory-of-difference-fields\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":99,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-277","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/posts\/277","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/users\/99"}],"replies":[{"embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/comments?post=277"}],"version-history":[{"count":2,"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/posts\/277\/revisions"}],"predecessor-version":[{"id":298,"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/posts\/277\/revisions\/298"}],"wp:attachment":[{"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/media?parent=277"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/categories?post=277"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/tags?post=277"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}