Read the Problem Statement and the Current (but still incomplete) Analysis below, then carry out all items in Tasks to Complete.
Write rigorously for a graduate audience; your final output must be self-contained.

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Problem Statement

Let \{V_n\}{n\ge1} be complex vector spaces with \dim{\Bbb C}V_n=m_n\to\infty.
Each V_n carries some norm \|\cdot\|_n.  Fix positive integers N,d.

For each n define k_n to be the smallest integer for which there exists a closed algebraic subvariety X_n\subset V_n such that
(1) X_n has at most N irreducible components;
(2) every component admits a Zariski-dense polynomial parametrisation \Phi:\Bbb C^{k_n}\to V_n whose coordinate functions are polynomials of degree \le d;
(3) (uniform density) there is a radius \varepsilon>0 independent of n with
\forall v\in V_n,\; \|v\|_n\le1 \implies \operatorname{dist}(v,X_n)\le\varepsilon .

Question. Must one have
\limsup_{n\to\infty}\frac{k_n}{m_n}=1\ ?

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Current Analysis (corrected to include the ambient-dimension factor)

Notation. Let M_n=2m_n and K_n=2k_n be the real dimensions, and set c_n’=M_n-K_n.
	1.	Packing lower bound.
For the Euclidean unit ball B_n\subset V_n one has
\mathcal N\!\bigl(B_n,\varepsilon\bigr)\;\ge\;\bigl(C_0/\varepsilon\bigr)^{M_n},
where \mathcal N(Z,\delta) is the minimal number of radius-\delta balls that cover Z.
	2.	Yomdin–Gromov covering bound (sharp form).
For every irreducible algebraic set Y\subset\Bbb R^{M_n} of degree \le d and intrinsic real dimension K_n,
\mathcal N(Y,\delta)\;\le\;\bigl(C_d\,M_n^{3/2}/\delta\bigr)^{K_n},
with C_d depending only on d.  M_n^{3/2} is essential in all known proofs.
	3.	Combining the bounds.
Since B_n\subset X_n\oplus B(0,\varepsilon),
\bigl(C_0/\varepsilon\bigr)^{M_n}
\;\le\;
N\bigl(C_d M_n^{3/2}/\varepsilon\bigr)^{K_n}.
Taking logs and writing c_n’=M_n-K_n,
c_n’\,\log\!\frac{C_0}{\varepsilon}
\;\le\;
\log N + K_n\Bigl[\tfrac32\log M_n + \log(C_d/C_0)\Bigr].
Dividing by M_n\log(C_0/\varepsilon) gives
\frac{c_n’}{M_n}
\;\le\;
\frac{\log N}{M_n\log(C_0/\varepsilon)}
+\frac{K_n}{M_n}\,
\frac{\tfrac32\log M_n + \log(C_d/C_0)}{\log(C_0/\varepsilon)}.
	4.	Consequences.
If \varepsilon=\varepsilon_n\downarrow0 with \log(1/\varepsilon_n)\gg\log M_n, the rightmost fraction tends to zero, forcing k_n/m_n\to1.
For a fixed \varepsilon>0 the denominator stays constant while \tfrac32\log M_n grows, so the inequality only gives a lower bound
\frac{k_n}{m_n}
\;\ge\;
\frac{\log(C_0/\varepsilon)}{\log(C_0/\varepsilon)+\tfrac32\log M_n+O(1)},
which neither tends to 1 nor rules out any positive lim-sup < 1.
Thus the original question is unresolved: we still need an explicit algebraic construction (or a rigorous citation) showing that for fixed \varepsilon>0 one can keep k_n/m_n bounded away from 1.

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Tasks to Complete
	1.	Provide an explicit algebraic family or a published theorem (with precise reference) that, for every fixed 0<\varepsilon<1 and integers N,d, produces varieties X_n\subset V_n\cong\Bbb C^{m_n} satisfying conditions (1)–(3) but with
\limsup_{n\to\infty}\frac{k_n}{m_n}<1.
The construction must respect the degree‐\le d constraint and use at most N components.
	2.	Assemble the final theorem.
Show the two-way statement:
	•	(i) If \varepsilon_n shrinks so that \log(1/\varepsilon_n)\gg\log m_n, then k_n/m_n\to1.
	•	(ii) For any fixed \varepsilon>0 there exist sequences with \limsup k_n/m_n<1.
Include a full, gap-free proof using the entropy inequality and your construction / citation.
	3.	Eliminate all previous sign or factor errors.
Track constants carefully, cite every external result precisely, and emphasise why a degree-d algebraic set cannot “space-fill’’ a high-dimensional ball unless its intrinsic dimension is large.

Return only the completed exposition (problem restatement, construction or citation, proofs, and final theorem).  Do not include planning notes.