--- title: Plus-Pure Threshold (PPT) in Mixed Characteristic url: https://www.emergentmind.com/topics/plus-pure-threshold-ppt type: topic --- # Plus-Pure Threshold (PPT) in Mixed Characteristic Searching arXiv for the most relevant papers on plus-pure thresholds in mixed characteristic. The **plus-pure threshold** is a mixed-characteristic singularity invariant defined for a complete Noetherian local domain of residue characteristic \(p>0\). It is intended as the mixed-characteristic analogue of the \(F\)-pure threshold in characteristic \(p\) and, more indirectly, of the log-canonical threshold in characteristic \(0\). In the regular complete local setting, it is defined by testing purity of the map \(R\xrightarrow{1\mapsto f^t}\widehat{R^+}\), or equivalently by membership of \(f^t\) in the maximal ideal times the absolute integral closure. Recent work has shown that this invariant is computable in nontrivial families, especially for cusp-like hypersurfaces of the form \(p^a+x^b\), and that it exhibits behavior both analogous to and distinct from its equal-characteristic counterparts [2501.07528], [2509.07217]. ## 1. Definition and formal framework Let \((S,\mathfrak m)\) be a complete Noetherian local domain of residue characteristic \(p>0\), and let \(0\neq f\in \mathfrak m\). The plus-pure threshold is defined by \[ \ppt(f):=\sup\Big\{\alpha\in \mathbf R_{\ge 0}\ \Big|\ S\xrightarrow{1\mapsto f^\alpha} S^+ \text{ is pure}\Big\}. \] This is the formulation given for complete local domains in mixed characteristic, with \(S^+\) denoting the absolute integral closure [2501.07528]. In the setting emphasized in later work, where \((R,m)\) is a complete Noetherian local domain of mixed characteristic \((0,p>0)\), the same invariant is written \[ \ppt(f\in R)\coloneqq \sup\{t\in \mathbb Q_{>0}\mid R\xrightarrow{1\mapsto f^t}\widehat{R^+}\text{ is pure}\}, \] using the \(p\)-adic completion \(\widehat{R^+}\) [2509.07217]. For a regular local ring, the definition admits a concrete ideal-membership reformulation. One has \[ \ppt(f)=\sup\{t\in \mathbb Q_{>0}\mid f^t\notin m\widehat{R^+}\} =\inf\{t\in \mathbb Q_{>0}\mid f^t\in mR^+\}. \] This criterion is central in explicit computations, because it reduces the threshold to deciding whether a rational power \(f^t\) lies in \(mR^+\) [2509.07217]. The earlier mixed-characteristic treatment also records equivalent conditions in the regular case: purity or splitting of the map to \(S^+\) or \(\widehat{S^+}\), splitting after passage to finite intermediate extensions, membership \(f^\alpha\notin \mathfrak m S^+\), and triviality of the plus-test ideal \(\tau_+(S,f^\alpha)=S\) [2501.07528]. This places the invariant inside the big Cohen–Macaulay and absolute integral closure framework. A plausible implication is that the plus-pure threshold is best viewed not merely as an analogue by formal resemblance, but as the first threshold detected by mixed-characteristic test-ideal methods. ## 2. Relation to \(F\)-pure and log-canonical thresholds A basic comparison established for cusp-like hypersurfaces is \[ \fpt(R_0,f_0)\le \ppt(R,f)\le \frac1a+\frac1b, \] for \[ R=\mathbf Z_p\llbracket x\rrbracket,\qquad f=p^a+x^b, \] and \[ R_0=\mathbf F_p\llbracket t,x\rrbracket,\qquad f_0=t^a+x^b. \] Here \(\frac1a+\frac1b\) is the expected log-canonical threshold in the corresponding characteristic-zero model, while \(\fpt(R_0,f_0)\) is the equal-characteristic \(p\) \(F\)-pure threshold [2501.07528]. More generally, in mixed-characteristic regular local rings one has the structural inequalities \[ \ppt(f)\le \lct(f),\qquad \ppt(f)\ge \fpt(\overline f), \] where \(\overline f\) is the reduction modulo the uniformizer [2509.07217]. These inequalities clarify the role of the invariant. It is bounded above by the birational threshold and below by the characteristic-\(p\) Frobenius threshold, but it is not determined by either one in general. This is not just a formal possibility: explicit examples show that \(\ppt\) can coincide with \(\fpt\), coincide with neither \(\fpt\) nor \(\lct\), and approach \(\fpt\) only after ramifying the coefficient DVR [2501.07528], [2509.07217]. The comparison with \(F\)-pure thresholds is especially strong for cusp-like singularities. If \[ \fpt(R_0,f_0)=\frac1a+\frac1b, \] then automatically \[ \ppt(R,f)=\frac1a+\frac1b}=\fpt(R_0,f_0) \] in that family [2501.07528]. By contrast, later work emphasizes that unramified mixed characteristic can force genuinely new behavior: \(\ppt\) may lie strictly between \(\fpt(\overline f)\) and \(\lct(f)\) [2509.07217]. ## 3. Cusp-like hypersurfaces \(p^a+x^b\) The first substantial computational body of work concerns hypersurfaces \[ f=p^a+x^b\in \mathbf Z_p\llbracket x\rrbracket. \] The main theorem states that, under explicit hypotheses, the mixed-characteristic threshold agrees with the equal-characteristic \(F\)-pure threshold of \[ f_0=t^a+x^b\in \mathbf F_p\llbracket t,x\rrbracket. \] Precisely, assume \[ \fpt(R_0,f_0)\neq \frac1a+\frac1b, \] choose \(e\ge 1\) such that \[ p^e\fpt(R_0,f_0)\in \mathbf Z, \] and suppose \[ \fpt(R_0,f_0)\ge \frac1a+\frac1b-\frac{1}{ap^e}. \] Then \[ \ppt(R,f)=\fpt(R_0,f_0). \] A stronger version replaces \(a\) by an integer \(c\ge a-\lfloor a/p\rfloor\) and uses the function \[ \rho_e(\lambda)=\frac{\lceil \lambda p^e\rceil-1}{p^e} \] to formulate the criterion [2501.07528]. This yields many explicit congruence-dependent families where equality holds. For \(2\le a,b\le 5\), the paper records the following cases. | \((a,b)\) | Congruence/prime conditions for \(\ppt=\fpt\) | |---|---| | \((2,2),(2,3),(2,4),(2,5)\) | all \(p\) | | \((3,2)\) | all \(p\neq 2\) | | \((3,3)\) | all \(p\neq 3\) | | \((3,4)\) | all \(p\neq 3\) with \(p\not\equiv 11\pmod{12}\) | | \((3,5)\) | all \(p\neq 3,5\) with \(p\not\equiv 14\pmod{15}\) | | \((4,2)\) | all \(p\) | | \((4,3)\) | all \(p\neq 3\) with \(p\not\equiv 11\pmod{12}\) | | \((4,4)\) | all \(p\neq 2\) with \(p\not\equiv 3\pmod 4\) | | \((4,5)\) | all \(p\neq 2\) with \(p\not\equiv 3,19\pmod{20}\) | | \((5,2)\) | all \(p\neq 2,5\) with \(p\not\equiv 7,9\pmod{10}\) | | \((5,3)\) | all \(p\neq 3,5\) with \(p\not\equiv 8,14\pmod{15}\) | | \((5,4)\) | all \(p\neq 2\) with \(p\not\equiv 3,19\pmod{20}\) | | \((5,5)\) | all \(p\neq 5\) with \(p\not\equiv 2,4\pmod 5\) | The logic behind these formulas is explicit. The paper imports known algorithms for \[ \fpt(t^a+x^b), \] often expressed via base-\(p\) expansions, and then checks whether the equal-characteristic threshold lies close enough to \(\frac1a+\frac1b\) to trigger the comparison theorem [2501.07528]. The case \(p^2+x^3\) illustrates the outcome particularly well. One has \[ \ppt(\mathbf Z_p\llbracket x\rrbracket,p^2+x^3)=\fpt(\mathbf F_p\llbracket y,x\rrbracket,y^2+x^3), \] and the equal-characteristic value is \[ \begin{cases} 1/2 & \text{if }p=2,\\ 2/3 & \text{if }p=3,\\ (5p-1)/6p & \text{if }p\equiv 5\pmod 6,\\ 5/6 & \text{if }p\equiv 1\pmod 6. \end{cases} \] This suggests that, at least for many cusp-like binomials, the mixed-characteristic threshold is governed with surprising accuracy by the mod-\(p\) Frobenius geometry [2501.07528]. ## 4. Ramification and the limit to the \(F\)-pure threshold A major later development is the discovery that plus-pure thresholds converge to \(F\)-pure thresholds under increasing ramification of the base DVR. Let \((V,\varpi)\) be a mixed characteristic complete DVR, let \[ R=V\llbracket x_2,\dots,x_n\rrbracket, \] and let \(f\in m\) reduce to \(\overline f\) modulo \(\varpi\). Then \[ \lim_{e\to\infty}\ppt\big(f\in V[\varpi^{1/p^e}]\llbracket x_2,\dots,x_n\rrbracket\big) = \fpt\big(\overline f\in V/(\varpi)\llbracket x_2,\dots,x_n\rrbracket\big). \] This is a sharp asymptotic statement: ramifying by adjoining deeper and deeper \(p\)-power roots of the uniformizer drives the mixed-characteristic threshold to the equal-characteristic \(p\) threshold [2509.07217]. Moreover, if the \(F\)-pure threshold has terminating base-\(p\) expansion, \[ \fpt(\overline f)=a/p^e, \] then equality occurs already at that finite ramification level: \[ \ppt\big(f\in V[\varpi^{1/p^e}]\llbracket x_2,\dots,x_n\rrbracket\big)=a/p^e. \] This finite-level stabilization provides exact formulas in many ramified situations [2509.07217]. The phenomenon is especially transparent for ramified diagonal equations. If \[ R_a=W(k)[p^{1/p^a}]\llbracket x_2,\dots,x_d\rrbracket,\qquad f_a=p^{d/p^a}+x_2^d+\cdots+x_d^d, \] and if \(p^s\le d\frac12 \] for every \(e>0\), while the limit remains \(\frac12\). This separates convergence from finite stabilization [2509.07217]. ## 5. Unramified behavior and departures from equal-characteristic analogies While cusp-like calculations often yield \(\ppt=\fpt\), later work shows that unramified mixed characteristic has threshold phenomena with no exact equal-characteristic analogue. A central theorem states that certain mixed-characteristic analogues of positive-characteristic extremal singularities do not attain the corresponding extremal value. For example, for \[ f = p^{p^{e} + 1} + x_2^{p^e + 1} + \dots + x_n^{p^e + 1} \] or \[ f = p^{p^{e} + 1} + x_2^{p^e}x_3 + x_3^{p^e}x_2 + x_4^{p^e + 1} + \dots + x_n^{p^e + 1} \] in \(W(k)\llbracket x_2,\dots,x_n\rrbracket\), the reduction mod \(p\) is an extremal singularity, but \[ \ppt(f)>\fpt(\overline f)=\frac{1}{p^e}. \] This shows that unramified mixed characteristic can force a threshold strictly above the mod-\(p\) extremal value [2509.07217]. A related example involves \[ f=p^{p^e}+x_2^{p^e}+\cdots+x_n^{p^e}. \] For odd \(p\), one proves \[ f^{1/p^e}\notin (p,x_2,\dots,x_n)B \] for any big Cohen–Macaulay \(R^+\)-algebra \(B\), hence \[ \ppt(f)>\frac{1}{p^e}=\fpt(f_0). \] This is a nonreduced-mod-\(p\) analogue of the same rigidity phenomenon [2509.07217]. Such results suggest that the unramified plus-pure threshold detects arithmetic constraints not visible in either \(\fpt\) or \(\lct\) alone. A plausible implication is that ramification is not just a technical device for computation; it separates two genuinely different regimes of mixed-characteristic singularity theory. ## 6. \(p\)-th roots modulo \(p^2\), upper bounds, and perfectoid purity Another structural theme is the effect of congruence to a \(p\)-th power. In an unramified regular local ring of mixed characteristic, if \(f\) admits a \(p\)-th root modulo \(p^2\), meaning \[ f\equiv h^p \pmod{p^2}, \] then one has the uniform bound \[ \ppt(f)\le 1-\frac1p. \] This follows from a more general theorem analyzing the extension \(S\to \overline{S[f^{1/p}]}\) and showing that suitable étaleness in codimension one forces the threshold below \(1\) [2509.07217]. This has a notable consequence for hypersurfaces of the form \[ f^p+p^2g. \] In a complete unramified regular local ring of mixed characteristic \(p>0\), such an equation never defines a perfectoid pure singularity, because \[ \ppt(f^p+p^2g)\le 1-\frac1p<1. \] Since \(\ppt(f)=1\) is equivalent in that setting to perfectoid purity of the hypersurface \(R/(f)\), the result shows that a full \(p\)-adic neighborhood of radius \(1/p^2\) around a \(p\)-th power consists of non-perfectoid-pure forms [2509.07217]. The same paper also proves a stronger upper bound in the presence of a primitive \(p\)-th root of unity. If \(f\) admits a \(p\)-th root modulo \((\zeta-1)^p\), then \[ \ppt(f)\le \frac1p. \] This is the tamely ramified analogue of the unramified \(1-1/p\) bound [2509.07217]. These results tie the plus-pure threshold to perfectoid purity in a concrete way. The threshold is not merely an abstract singularity exponent; it controls whether hypersurfaces are perfectoid pure, and it is highly sensitive to congruence patterns of the defining equation. ## 7. Elliptic and cubic examples The paper on ramified regular rings also studies hypersurfaces related to elliptic curves. For \[ f=p^3+x^3+y^3\in W(k)\llbracket x,y\rrbracket \] with \(p\equiv_3 2\), one proves \[ \ppt(f)\le 1-\frac1{p^2}. \] Combined with the general lower bound \(\ppt(f)\ge \fpt(\overline f)\), this yields \[ \ppt(p^3+x^3+y^3)\in \left[1-\frac1p,\ 1-\frac1{p^2}\right]. \] For \(p=2\), this sharpens to \[ \ppt(x^3+y^3+2^3)\in \left(\frac12,\frac34\right]. \] Since the characteristic-\(2\) reduction satisfies \[ \fpt(x^3+y^3+z^3)=\frac12 \] and the log canonical threshold is \[ \lct=1, \] this gives a concrete example where \[ \frac12<\ppt<1. \] Thus the plus-pure threshold is neither the corresponding \(F\)-pure threshold nor the log-canonical threshold [2509.07217]. A ramified counterpart behaves more simply. If \(R=V\llbracket x,y,z\rrbracket\) with \(V\) containing a \(p\)-th root of \(p\), and \(\overline f\) is a homogeneous cubic defining a nonsingular elliptic curve \(E\), then \[ \ppt(f)=\fpt(\overline f)= \begin{cases} 1 & \text{if \(E\) is ordinary},\\ 1-\dfrac1p & \text{if \(E\) is supersingular}. \end{cases} \] This is another instance of the general principle that sufficient ramification collapses the mixed-characteristic threshold to the familiar characteristic-\(p\) value [2509.07217]. Earlier work also gave the sporadic example \[ f=x^2+y^3\in \mathbf Z_2\llbracket x,y\rrbracket, \] for which \[ \ppt(R,f)>1/2=\fpt(\mathbf F_2\llbracket x,y\rrbracket,f). \] The exact value was left open. This was among the first clear demonstrations that \(\ppt\) can exceed \(\fpt\) even for simple cusp-like forms [2501.07528]. ## 8. Methods of computation The computational methods used across these papers are remarkably concrete. One basic device is the ideal-membership reformulation \[ \ppt(f)=\sup\{t\mid f^t\notin mR^+\}, \] which turns threshold calculations into containment problems in absolute integral closures or big Cohen–Macaulay algebras [2501.07528], [2509.07217]. Another method is the use of perfectoid-style containment lemmas. If \(\epsilon\in(0,1]\), then one has statements of the form \[ z\in (p^\epsilon,y_1,\dots,y_s) \iff z^{1/p^e}\in (p^{\epsilon/p^e},y_1^{1/p^e},\dots,y_s^{1/p^e}), \] which serve as mixed-characteristic substitutes for Frobenius manipulations [2501.07528], [2509.07217]. These are especially important in passing from equal-characteristic containments to mixed-characteristic ones after adjoining \(p\)-power roots of the uniformizer. Blow-up geometry also enters. For cusp-like equations \(u^{a\ell}+x^{b\ell}\), the normalized blowup yields an exceptional divisor \(E\cong \mathbf P^1_k\), and one computes \[ K_{X/R}=(a+b-1)E,\qquad \pi^*\Div(f)=D+(\ell ab)E, \] with different \[ \Diff_E=\left(1-\frac1a\right)P_1+\left(1-\frac1b\right)P_2. \] These formulas produce the upper bound \(\ppt\le \frac1a+\frac1b\) and allow adjunction arguments that relate \(\ppt\) to \(F\)-regularity on the exceptional divisor [2501.07528]. A further ingredient is the finite-cover or cyclic-cover principle \[ (R, u^{1-1/a}x^{1-1/b}(u^\ell+x^\ell)^t)\text{ is \(F\)-regular} \iff (R,(u^{a\ell}+x^{b\ell})^t)\text{ is \(F\)-regular,} \] which links mixed-characteristic geometry to equal-characteristic binomial \(F\)-singularity computations [2501.07528]. Finally, later work uses valuation and divisibility arguments, including Kummer-type valuations of binomial coefficients, to prove upper bounds such as \[ \ppt(p^3+x^3+y^3)\le 1-\frac1{p^2}. \] This suggests that plus-pure threshold computations sit at an unusual intersection of perfectoid algebra, valuation theory, and explicit arithmetic combinatorics [2509.07217]. ## 9. Conceptual significance and open directions The plus-pure threshold has emerged as the mixed-characteristic threshold analogue of the \(F\)-pure threshold, but it is not merely a transplanted equal-characteristic invariant. Several facts support this assessment. First, it behaves functorially with respect to ramification: \[ \ppt \to \fpt \] under adjoining deep \(p\)-power roots of the uniformizer [2509.07217]. Second, in many low-complexity cusp-like families one has exact equality \[ \ppt=\fpt, \] often with explicit congruence conditions on \(p\) [2501.07528]. Third, in unramified settings it can sit strictly between \(\fpt\) and \(\lct\), and it can fail to realize extremal mod-\(p\) values [2509.07217]. This combination of approximation and divergence is distinctive. Several open problems remain explicit in the literature. One is whether, for binomials \(p^a+x^b\), the only possible values of \(\ppt\) are \[ \fpt(f_0)\quad\text{or}\quad \frac1a+\frac1b. \] This is posed directly after the cusp-like computations [2501.07528]. Another is to determine unresolved small cases such as \[ \ppt(\mathbf Z_3\llbracket x\rrbracket,x^3+27),\qquad \ppt(\mathbf Z_2\llbracket x\rrbracket,x^4+16). \] Later work also raises questions about when the ramification limit theorem stabilizes at finite level, and how far characteristic-\(p\) jumping-number phenomena extend to mixed characteristic [2509.07217]. A further distinction from characteristic \(p\) is that \(p\cdot \ppt(f)\) need not be a jumping number. The \(2\)-adic cubic example supplies this directly, in contrast with the characteristic-\(p\) theory of \(F\)-jumping numbers [2509.07217]. This suggests that the threshold theory attached to \(\widehat{R^+}\) is structurally richer than a direct Frobenius analogue. ## 10. Summary The plus-pure threshold is defined for mixed-characteristic complete local domains by \[ \ppt(f)=\sup\{t\mid R\xrightarrow{1\mapsto f^t}\widehat{R^+}\text{ is pure}\}, \] and, in regular local rings, equivalently by \[ \ppt(f)=\sup\{t\mid f^t\notin m\widehat{R^+}\}. \] It is bounded by \[ \fpt(\overline f)\le \ppt(f)\le \lct(f), \] but its actual value is often subtler than either endpoint [2501.07528], [2509.07217]. For cusp-like singularities \(p^a+x^b\), extensive computations show that \(\ppt\) frequently agrees with the mod-\(p\) \(F\)-pure threshold \( \fpt(t^a+x^b)\), and explicit congruence conditions on \(p\) can force this equality [2501.07528]. Under increasing ramification of the coefficient DVR, one always has \[ \ppt \to \fpt, \] and sometimes equality occurs at finite ramification level [2509.07217]. By contrast, in unramified mixed characteristic, the invariant exhibits strictly new phenomena: extremal mod-\(p\) values may not be attained, forms congruent to \(p\)-th powers modulo \(p^2\) satisfy upper bounds such as \(\ppt\le 1-1/p\), and elliptic-type examples show \(\ppt\) can lie strictly between \(\fpt\) and \(\lct\) [2509.07217]. This suggests that the plus-pure threshold is best understood as a genuinely mixed-characteristic singularity invariant, controlled by absolute integral closure and perfectoid big Cohen–Macaulay methods, related to but not reducible to the familiar thresholds of equal characteristic. Source: https://www.emergentmind.com/topics/plus-pure-threshold-ppt