Aggressive geometry – Temporal complexity of a quantified linear program without free variables with only existential quantifications

All known examples of lower double exponential bounds for real quantizer elimination without free variables (essentially a decision problem) involve polynomial inequalities with degree $> 1 $

Is there an example of double exponentiation with polyhedral inequalities where polyhedral inequalities relate to degree? $ 1 $ for the case of non-free variables?

We know $ exists x in mathbb R ^ n: Axis leq b $ is a standard linear program.

I mainly look at the following case of quantized linear program without any free variable with only existential quantifications.

Can we still solve programs like $$ exists x_1 in mathbb R ^ {n_1} points exists x_t in mathbb R ^ {n_t} $$$$ A_1 (x_1, x_2) ^ T leq b_1 coin dots coin A_ {t-1} (x_ {t-1}, x_t) ^ T leq b_ {t-1} $$
and
$$ exists x_1 in mathbb R ^ {n_1} points exists x_t in mathbb R ^ {n_t} $$$$ A_1 (x_1, x_2) ^ T leq b_1 corner A_2 (x_1, x_2, x_3) ^ T leq b_2 coin dots corner A_ {t-1} (x_1, points, x_ {t- 1}, x_t) ^ T leq b_ {t-1} $$
at the hour $ mathsf {poly} (t, sum_ {i = 1} ^ t (m_i + n_i)) $ or $ A_i $ are matrices in $ mathbb R ^ {m_i times n_i} $ and $ b_i $ are vectors in $ mathbb R ^ {m_i} $ by collapse quantifications into one big $ exists $ quantify and consider as a standard linear program and solve through standard linear programming techniques?

Or is it possible that there are scenarios in which the program would require time $$ big ( sum_ {i = 1} ^ t (m_i + n_i) big) ^ {2 ^ {O (t)}} mathsf {poly} (t, sum_ {i = 1} ^ t (m_i + n_i)) $$ or maybe as much as $$ 2 ^ {O big ( big ( sum_ {i = 1} ^ t (m_i + n_i) big) ^ t big)} mathsf {poly} (t, sum_ {i = 1} t (m_i + n_i))? $$

Related, connected, related: Elimination of the quantizer without free variables and without polyhedral inequalities

In this blog introductory article, https://cstheory.blogoverflow.com/2011/11/something-you-should-know-about-quantifier-elimination-part-i/, it is mentioned in the last line that "I do not know if a doubly exponential lower bound is known for the decision problem when there are no free variables".

I want to know if this comment applies to scenarios without alternation.

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Aggressive geometry – Pushing ahead of locally free sheaves and solving singularities

Let $ X $ to be a Noetherian, affine, isolated of dimension at least $ 2 $ and $ pi: widetilde {X} to X $ a resolution of singularities. Let $ mathcal {E} $ to be a locally free sheaf on $ widetilde {X} $ such as $ pi _ * mathcal {E} $ is free $ mathcal {O} _X $-module. is $ mathcal {E} $ necessarily trivial, that is to say a direct sum of copies of $ mathcal {O} _ { widetilde {X}} $?