Zariski's multiplicity conjecture: does embedded topology determine the multiplicity of a hypersurface singularity?

Two independent counterexample constructions. September 24: for some finite $N>3$ divisible by eight, reduced weighted homogeneous real polynomials $f_1,f_2$ on $\mathbb{C}^N$ with isolated critical points, $\mathrm{ord}_0f_1=2$, $\mathrm{ord}_0f_2=3$, and $(\mathbb{C}^N,V(f_1),0)\cong(\mathbb{C}^N,V(f_2),0)$; the germs are also topologically right equivalent, so the right-equivalence version and its mod-2 version (Sampaio 2025, Conjectures 2 and Z mod 2) fail too. September 27: reduced convergent germs $f_4,f_5$ on $\mathbb{C}^4$ with isolated critical points, orders $4$ and $5$, ambiently homeomorphic zero sets. Not addressed: hypersurfaces in $\mathbb{C}^3$, and metric (bi-Lipschitz) or family versions, where positive results stand.

Two independent counterexample constructions. September 24: for some finite $N>3$ divisible by eight, reduced weighted homogeneous real polynomials $f_1,f_2$ on $\mathbb{C}^N$ with isolated critical points, $\mathrm{ord}_0f_1=2$, $\mathrm{ord}_0f_2=3$, and $(\mathbb{C}^N,V(f_1),0)\cong(\mathbb{C}^N,V(f_2),0)$; the germs are also topologically right equivalent, so the right-equivalence version and its mod-2 version (Sampaio 2025, Conjectures 2 and Z mod 2) fail too. September 27: reduced convergent germs $f_4,f_5$ on $\mathbb{C}^4$ with isolated critical points, orders $4$ and $5$, ambiently homeomorphic zero sets. Not addressed: hypersurfaces in $\mathbb{C}^3$, and metric (bi-Lipschitz) or family versions, where positive results stand.