You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'Let $X\subset {\mathbb{C}}^n$ be a smooth irreducible affine variety of dimension $k$ and let $F: X\to {\mathbb{C}}^m$ be a polynomial mapping. We prove that if $m\ge k$, then there is a Zariski open dense subset $U$ in the space of linear mappings ${\mathcal L}({\mathbb{C}}^n,{\mathbb{C}}^m)$ such that for every $L\in U$ the mapping $F+L$ is a finite mapping. Moreover, we can choose $U$ in this way, that all mappings $F+L; L\in U$ are topologically equivalent.'
address: 'Zbigniew Jelonek - Instytut Matematyczny, Polska Akademia Nauk, Śniadeckich 8, 00-956 Warszawa, Poland.'
author:
- Zbigniew Jelonek
title: On finite polynomial mappings
---

Introduction
============

Assume that we have an algebraic family $\mathcal F$ of polynomial generically-finite mappings $f_p: X\to {\mathbb{C}}^m; \ p\in {\mathcal F}$, where $X$ is a smooth irreducible affine variety. It is important to know the behavior of proper mappings in a such family. In general, proper mappings does not form an algebraic subset of $\mathcal F$, but only constructible one. However we show in this note that we have some regular behavior in such family.

As an application we show that if $X\subset {\mathbb{C}}^n$ is a smooth irreducible affine variety of dimension $k$ and let $F: X\to {\mathbb{C}}^m$ be a polynomial mapping. If $m\ge k$, then there exists a Zariski open dense subset $U$ in the space of linear mappings ${\mathcal L}({\mathbb{C}}^n,{\mathbb{C}}^m)$ such that:

a\) for every $L\in U$ the mapping $F+L$ is a finite mapping.

b\) all mappings $F+L, L\in U$ are topologically equivalent.

Let us recall that mappings $f,g: X\to Y$ are topologically equivalent, if there exist homeomorphisms $\phi: X\to X$ and $\psi: Y\to Y$ such that $f=\psi\circ g\circ \phi.$

Main results
============

Let us start with the following:

\[rodzina\] Let $P, X, Y$ be smooth irreducible affine algebraic varieties and let $F: P\times X\to P\times Y$ be a generically finite mapping. The mapping $F$ induces a family ${\mathcal F}=\{ f_p(\cdot)=F(p,\cdot), \ p\in P \}.$ Then either there exists a Zariski open dense subset $U\subset P$ such that for every $p\in P$ a mapping $f_p$ is proper, or there exists a Zariski open dense subset $V\subset P$ such that for every $p\in P$ a mapping $f_p$ is not proper.

In the first case we have:

a\) foe every non-proper mappings $f_p$ in the family $\mathcal F$ we have $\mu(f_p)<\mu(F)$, where $\mu(f)$ denotes the geometric degree of $f,$

b\) all generic mappings $f_p$ are topologically equivalent, i.e., there exists a Zariski open dense subset $W\subset P$, such that for every $p,q\in W$ mappings $f_p$ and $f_q$ are topologically equivalent.

First of all note that for every $(p,x)\in P\times X$ we have $\mu_{(p,x)}(F)= \mu_x (f_p)$ (here $\mu_x(f)$ denotes the local multiplicity of $f$ in $x$). In the sequel we use the fact that a mapping $g: X\to Y$ is proper over a point $y\in Y$ if and only if $\sum_{g(x)=y} \mu_x(g)=\mu(g)$ (see [@Je], [@Je1]).

Let $S$ be the non-properness set of $F$ (see e.g. [@Je], [@Je1]). If $S=\emptyset$, then all mappings $f_p$ are proper. Hence we can assume that $S\not= \emptyset.$ Consider the canonical projection $\pi: S\to P.$ We have two possibilities:

[(1)]{} $\pi(S)$ is dense in $P.$

[(2)]{} $\pi(S)$ is not dense in $P.$

In the case a) a generic mapping $f_p$ is not proper. In the second case note that $S$ has dimension dim $P$ + dim $X -1$ and the fiber of $\pi$ has dimension at most dim $X.$ This immediately implies that the set $\overline{\pi(S)}$ is a hypersurface in $M$. Moreover, fibers of $\pi$ are the whole space $X.$ This means that for all $p\in \pi(S)$ we have $\mu(f_p)<\mu(F).$ Of course outside $\pi(S)$ mappings $f_p$ are proper. Two such a generic mappings are topologically equivalent by [@Je2], Theorem 4.3.

Now we state our main result:

Let $X\subset {\mathbb{C}}^n$ be a smooth irreducible affine variety of dimension $k$ and let $F: X\to {\mathbb{C}}^m$ be a polynomial mapping. If $m\ge k$, then there existss a Zariski open dense subset $U$ in the space of linear mappings ${\mathcal L}({\mathbb{C}}^n,{\mathbb{C}}^m)$ such that:

a\) for every $L\in U$ the mapping $F+L$ is a finite mapping.

b\) all mappings $F+L, L\in U$ are topologically equivalent.

Let $G: X\ni x \mapsto (x, F(x))\in X\times {\mathbb{C}}^m$ and $\tilde{X}=graph(G)\cong X.$ Since $m\ge \dim \tilde{X}$ a generic linear projection $\pi : \tilde{X}\to {\mathbb{C}}^m$ is a proper mapping. Hence also the mapping $\pi\circ G$ is proper. Consequently we get that for a general matrix $A\in GL(m,m)$ and general linear mapping $L\in {\mathcal L}({\mathbb{C}}^n,{\mathbb{C}}^m)$ the mapping $H(A,L)=A(F_1,...,F_m)^T+L$ is proper. Hence also the mapping $A^{-1}\circ H(A,L)$ is proper. This means that the mapping $F+ A^{-1}(l_1,...,l_m)^T$ (where $L=(l_1,...,l_m)$) is proper. But we can specialize the matrix $A$ to the identity and the mapping $L$ to a given linear mapping $L_0\in {\mathcal L}({\mathbb{C}}^n,{\mathbb{C}}^m)\}$. Hence we see that there is at least dense subset of linear mappings $L\in {\mathcal L}({\mathbb{C}}^n,{\mathbb{C}}^m)$ such that the mapping $F+L: X\to {\mathbb{C}}^m$ is proper. Consider the algebraic family ${\mathcal F}=\{ F+L, L\in {\mathcal L}({\mathbb{C}}^n,{\mathbb{C}}^m)\}$. By Theorem \[rodzina\] we see that there exists a Zariski dense open subset $U\subset {\mathcal L}({\mathbb{C}}^n,{\mathbb{C}}^m)$ such that every mapping $F+L; \ L\in U$ is proper and all these mappings are topologically equivalent.

[99]{}

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