A common-SNC carry formula for the negative-perturbation subadditivity question

Definitions and mathematical statement

Takagi's recovered question is: let $(R,\mathfrak m)$ be a regular local ring of characteristic $p$, with ideals $I,\mathfrak a,\mathfrak b$ satisfying \[ \mathfrak a\subset I,\qquad \mathcal J(I^{-\epsilon}\mathfrak a^s)\subset\mathfrak m,\qquad \mathcal J(\mathfrak b^t)\subset\mathfrak m. \] Does one have \[ \mathcal J(I^{-\epsilon}\mathfrak a^s\mathfrak b^t) \subseteq\mathcal J(I^{-\epsilon}\mathfrak a^s)\mathcal J(\mathfrak b^t)? \] The calligraphic $\mathcal J$, admissible exponents, and meaning of the negative perturbation $I^{-\epsilon}$ are not defined in the source. This is a formal decorated-ideal expression, not an ordinary negative power of a possibly noninvertible ideal. In particular a characteristic-zero multiplier ideal cannot silently be substituted for an unspecified characteristic-$p$ invariant.

The report gives an explicitly restricted common-simple-normal-crossings interpretation. Let $x_1,\ldots,x_r$ belong to one regular parameter system and take nonzero principal monomial ideals \[ I=(\prod_i x_i^{\iota_i}),\quad \mathfrak a=(\prod_i x_i^{\alpha_i}),\quad \mathfrak b=(\prod_i x_i^{\beta_i}), \qquad \iota_i,\alpha_i,\beta_i\in{\mathbb{Z}}_{\geq0}, \] with $\alpha_i\geq\iota_i$, expressing $\mathfrak a\subseteq I$. In the nonnegative-exponent regime put \[ c_i=s\alpha_i-\epsilon\iota_i\geq0,\qquad t\geq0. \] The report notes $0<\epsilon<s$ is sufficient for these coefficient inequalities. For the characteristic-zero SNC multiplier ideal, or for the characteristic-$p$ SNC generalized test ideal under the standard $F$-finite regular hypotheses, it proposes \[ \mathcal J(I^{-\epsilon}\mathfrak a^s) =(\prod_i x_i^{\lfloor c_i\rfloor}),\qquad \mathcal J(\mathfrak b^t)=(\prod_i x_i^{\lfloor t\beta_i\rfloor}), \] \[ \mathcal J(I^{-\epsilon}\mathfrak a^s\mathfrak b^t) =(\prod_i x_i^{\delta_i})\, \mathcal J(I^{-\epsilon}\mathfrak a^s)\mathcal J(\mathfrak b^t), \qquad \delta_i=\left\lfloor\{c_i\}+\{t\beta_i\}\right\rfloor\in\{0,1\}. \] Braces denote fractional parts, so the correction is a squarefree carry monomial. The claimed inclusion follows in this interpretation, with equality exactly when every $\delta_i=0$, including the boundary rule that a fractional-part sum equal to one gives a carry. This supplies a special case of the original proper-factor question when its two $\mathfrak m$-containments hold; it is not a definition of the source's $\mathcal J$ in general. The report includes all nonzero ideals in a DVR under the same exponent regime, but does not resolve the non-SNC case or negative combined coefficients.

Short English statement

The source asks a subadditivity question involving a negative ideal perturbation but leaves its invariant undefined. The report gives a common-SNC principal-monomial case with an exact squarefree fractional-carry formula, retaining the original properness conditions separately.

Sources

• AIM workshop "Numerical invariants of singularities and higher-dimensional algebraic varieties," item 5, attributed to Takagi: https://aimath.org/WWN/singularvariety/problems_final.pdf. No PDF or underlying TeX was accessed here.
• Local canonical registry aim-algebraic-geometry-notes.json, notes[314], tag problem; permanent UnsolvedMath ID $20000315$, source code catalogue entry. Original has the exponent control-character artifact U+000F, lost superscripts, and a trailing page number. The supplied clean statement recovers $-\epsilon,s,t$; the verification field claims underlying-TeX confirmation. Original remarks and literature field are empty.
• Supplied research attempt 1 proposing the common-SNC carry formula and DVR special case. No authoritative full bibliographic references or definition of the original $\mathcal J$ are supplied. The report is machine generated, not peer reviewed, and requires independent verification before citation.