There is a purely algebraic characterisation of Condition (B) due to Le and Teissier, see Proposition 1.3.8 of the paper

Lê Dũng Tráng; Teissier, Bernard, Limites d’espaces tangents en géométrie analytique. (Limits of tangent spaces in analytic geometry), Comment. Math. Helv. 63, No. 4, 540-578 (1988). ZBL0658.32010..

Here's a summary. Assume that $X$ is a complex algebraic variety (embedded in $\mathbb{C}^n$ if affine or $\mathbb{P}^n$ if projective) and that $Y \subset X$ is a smooth quasiprojective subvariety. Then the pair $(X_\text{reg},Y)$ satisfies Condition (B) if and only if there is a containment of ideals $$I[\textbf{Con}(X) \cap \textbf{Con}(Y)] \subset \overline{I}[\kappa_X^{-1}(Y)]$$

Some explanation: here $I[Z]$ means the generating ideal of $Z$, while $\textbf{Con}(X)$ is the conormal variety of $X$ and $\kappa_X:\textbf{Con}(X) \to X$ is the conormal map. The bar on the right side here denotes integral closure. I learned about all this from Chapter 4 of Flores and Teissier's amazing survey

Flores, Arturo Giles; Teissier, Bernard, Local polar varieties in the geometric study of singularities, Ann. Fac. Sci. Toulouse, Math. (6) 27, No. 4, 679-775 (2018). ZBL1409.14002.

Martin Helmer and I have a recent paper which uses this Le-Teissier criterion to algorithmically construct Whitney stratifications of complex varieties. The hard part is bypassing the integral closure, which is computationally prohibitive. Martin even has a Macaulay2 implementation on his webpage which you can play around with if you have actual example varieties to stratify :)

There is a purely algebraic characterisation of Condition (B) due to Le and Teissier, see Proposition 1.3.8 of the paper

Lê Dũng Tráng; Teissier, Bernard, Limites d’espaces tangents en géométrie analytique. (Limits of tangent spaces in analytic geometry), Comment. Math. Helv. 63, No. 4, 540-578 (1988). ZBL0658.32010..

Here's a summary. Assume that $X$ is a complex algebraic variety (embedded in $\mathbb{C}^n$ if affine or $\mathbb{P}^n$ if projective) and that $Y \subset X$ is a smooth quasiprojective subvariety. Then the pair $(X_\text{reg},Y)$ satisfies Condition (B) if and only if there is a containment of ideals $$I[\textbf{Con}(X) \cap \textbf{Con}(Y)] \subset \overline{I}[\kappa_X^{-1}(Y)]$$

Some explanation: here $I[Z]$ means the generating ideal of $Z$, while $\textbf{Con}(X)$ is the conormal variety of $X$ and $\kappa_X:\textbf{Con}(X) \to X$ is the conormal map. The bar on the right side here denotes integral closure. I learned about all this from Chapter 4 of Flores and Teissier's amazing survey

Flores, Arturo Giles; Teissier, Bernard, Local polar varieties in the geometric study of singularities, Ann. Fac. Sci. Toulouse, Math. (6) 27, No. 4, 679-775 (2018). ZBL1409.14002.

Martin Helmer and I have a recent paper which uses this Le-Teissier criterion to algorithmically construct Whitney stratifications of complex varieties. The hard part is bypassing the integral closure, which is computationally prohibitive. Martin even has a Macaulay2 implementation on his webpage which you can play around with if you have actual example varieties to stratify :)

Update (11th July 2023) There is an error in my paper with Martin mentioned above, which has been described and corrected in this erratum; the fix involves using primary decompositions rather than saturations. The arxiv version incorporates all of the corrections.

There is a purely algebraic characterisation of Condition (B) due to Le and Teissier, see Proposition 1.3.8 of the paper

Lê Dũng Tráng; Teissier, Bernard, Limites d’espaces tangents en géométrie analytique. (Limits of tangent spaces in analytic geometry), Comment. Math. Helv. 63, No. 4, 540-578 (1988). ZBL0658.32010..

Here's a summary. Assume that $X$ is a complex algebraic variety and $Y \subset X$ is a smooth quasiprojective subvariety. Then the pair $(X_\text{reg},Y)$ satisfies Condition (B) if and only if there is a containment of ideals $$I[\textbf{Con}(X) \cap \textbf{Con}(Y)] \subset \overline{I}[\kappa_X^{-1}(Y)]$$

Some explanation: here $I[Z]$ means the generating ideal of $Z$, while $\textbf{Con}(X)$ is the conormal variety of $X$ and $\kappa_X:\textbf{Con}(X) \to X$ is the conormal map. The bar on the right side here denotes integral closure. I learned about all this from Chapter 4 of Flores and Teissier's amazing survey

Flores, Arturo Giles; Teissier, Bernard, Local polar varieties in the geometric study of singularities, Ann. Fac. Sci. Toulouse, Math. (6) 27, No. 4, 679-775 (2018). ZBL1409.14002.

Martin Helmer and I have a recent paper which uses this Le-Teissier criterion to algorithmically construct Whitney stratifications of complex varieties. The hard part is bypassing the integral closure, which is computationally prohibitive. Martin even has a Macaulay2 implementation on his webpage which you can play around with if you have actual example varieties to stratify :)

There is a purely algebraic characterisation of Condition (B) due to Le and Teissier, see Proposition 1.3.8 of the paper

Lê Dũng Tráng; Teissier, Bernard, Limites d’espaces tangents en géométrie analytique. (Limits of tangent spaces in analytic geometry), Comment. Math. Helv. 63, No. 4, 540-578 (1988). ZBL0658.32010..

Here's a summary. Assume that $X$ is a complex algebraic variety (embedded in $\mathbb{C}^n$ if affine or $\mathbb{P}^n$ if projective) and that $Y \subset X$ is a smooth quasiprojective subvariety. Then the pair $(X_\text{reg},Y)$ satisfies Condition (B) if and only if there is a containment of ideals $$I[\textbf{Con}(X) \cap \textbf{Con}(Y)] \subset \overline{I}[\kappa_X^{-1}(Y)]$$

Some explanation: here $I[Z]$ means the generating ideal of $Z$, while $\textbf{Con}(X)$ is the conormal variety of $X$ and $\kappa_X:\textbf{Con}(X) \to X$ is the conormal map. The bar on the right side here denotes integral closure. I learned about all this from Chapter 4 of Flores and Teissier's amazing survey

Flores, Arturo Giles; Teissier, Bernard, Local polar varieties in the geometric study of singularities, Ann. Fac. Sci. Toulouse, Math. (6) 27, No. 4, 679-775 (2018). ZBL1409.14002.

Martin Helmer and I have a recent paper which uses this Le-Teissier criterion to algorithmically construct Whitney stratifications of complex varieties. The hard part is bypassing the integral closure, which is computationally prohibitive. Martin even has a Macaulay2 implementation on his webpage which you can play around with if you have actual example varieties to stratify :)