More on Marketing

Francesca Furchtgott
3 years ago
Giving customers what they want or betraying the values of the brand?
A J.Crew collaboration for fashion label Eveliina Vintage is not a paradox; it is a solution.
Eveliina Vintage's capsule collection debuted yesterday at J.Crew. This J.Crew partnership stopped me in my tracks.
Eveliina Vintage sells vintage goods. Eeva Musacchia founded the shop in Finland in the 1970s. It's recognized for its one-of-a-kind slip dresses from the 1930s and 1940s.
I wondered why a vintage brand would partner with a mass shop. Fast fashion against vintage shopping? Will Eveliina Vintages customers be turned off?
But Eveliina Vintages customers don't care about sustainability. They want Eveliina's Instagram look. Eveliina Vintage collaborated with J.Crew to give customers what they wanted: more Eveliina at a lower price.
Vintage: A Fashion Option That Is Eco-Conscious
Secondhand shopping is a trendy response to quick fashion. J.Crew releases hundreds of styles annually. Waste and environmental damage have been criticized. A pair of jeans requires 1,800 gallons of water. J.Crew's limited-time deals promote more purchases. J.Crew items are likely among those Americans wear 7 times before discarding.
Consumers and designers have emphasized sustainability in recent years. Stella McCartney and Eileen Fisher are popular eco-friendly brands. They've also flocked to ThredUp and similar sites.
Gap, Levis, and Allbirds have listened to consumer requests. They promote recycling, ethical sourcing, and secondhand shopping.
Secondhand shoppers feel good about reusing and recycling clothing that might have ended up in a landfill.
Eco-conscious fashionistas shop vintage. These shoppers enjoy the thrill of the hunt (that limited-edition Chanel bag!) and showing off a unique piece (nobody will have my look!). They also reduce their environmental impact.
Is Eveliina Vintage capitalizing on an aesthetic or is it a sustainable brand?
Eveliina Vintage emphasizes environmental responsibility. Vogue's Amanda Musacchia emphasized sustainability. Amanda, founder Eeva's daughter, is a company leader.
But Eveliina's press message doesn't address sustainability, unlike Instagram. Scarcity and fame rule.
Eveliina Vintages Instagram has see-through dresses and lace-trimmed slip dresses. Celebrities and influencers are often photographed in Eveliina's apparel, which has 53,000+ followers. Vogue appreciates Eveliina's style. Multiple publications discuss Alexa Chung's Eveliina dress.
Eveliina Vintage markets its one-of-a-kind goods. It teases future content, encouraging visitors to return. Scarcity drives demand and raises clothing prices. One dress is $1,600+, but most are $500-$1,000.
The catch: Eveliina can't monetize its expanding popularity due to exorbitant prices and limited quantity. Why?
Most people struggle to pay for their clothing. But Eveliina Vintage lacks those more affordable entry-level products, in contrast to other luxury labels that sell accessories or perfume.
Many people have trouble fitting into their clothing. The bodies of most women in the past were different from those for which vintage clothing was designed. Each Eveliina dress's specific measurements are mentioned alongside it. Be careful, you can fall in love with an ill-fitting dress.
No matter how many people can afford it and fit into it, there is only one item to sell. To get the item before someone else does, those people must be on the Eveliina Vintage website as soon as it becomes available.
A Way for Eveliina Vintage to Make Money (and Expand) with J.Crew Its following
Eveliina Vintages' cooperation with J.Crew makes commercial sense.
This partnership spreads Eveliina's style. Slightly better pricing The $390 outfits have multicolored slips and gauzy cotton gowns. Sizes range from 00 to 24, which is wider than vintage racks.
Eveliina Vintage customers like the combination. Excited comments flood the brand's Instagram launch post. Nobody is mocking the 50-year-old vintage brand's fast-fashion partnership.
Vintage may be a sustainable fashion trend, but that's not why Eveliina's clients love the brand. They only care about the old look.
And that is a tale as old as fashion.

M.G. Siegler
3 years ago
Apple: Showing Ads on Your iPhone
This report from Mark Gurman has stuck with me:
In the News and Stocks apps, the display ads are no different than what you might get on an ad-supported website. In the App Store, the ads are for actual apps, which are probably more useful for Apple users than mortgage rates. Some people may resent Apple putting ads in the News and Stocks apps. After all, the iPhone is supposed to be a premium device. Let’s say you shelled out $1,000 or more to buy one, do you want to feel like Apple is squeezing more money out of you just to use its standard features? Now, a portion of ad revenue from the News app’s Today tab goes to publishers, but it’s not clear how much. Apple also lets publishers advertise within their stories and keep the vast majority of that money. Surprisingly, Today ads also appear if you subscribe to News+ for $10 per month (though it’s a smaller number).
I use Apple News often. It's a good general news catch-up tool, like Twitter without the BS. Customized notifications are helpful. Fast and lovely. Except for advertisements. I have Apple One, which includes News+, and while I understand why the magazines still have brand ads, it's ridiculous to me that Apple enables web publishers to introduce awful ads into this experience. Apple's junky commercials are ridiculous.
We know publishers want and probably requested this. Let's keep Apple News ad-free for the much smaller percentage of paid users, and here's your portion. (Same with Stocks, which is more sillier.)
Paid app placement in the App Store is a wonderful approach for developers to find new users (though far too many of those ads are trying to trick users, in my opinion).
Apple is also planning to increase ads in its Maps app. This sounds like Google Maps, and I don't like it. I never find these relevant, and they clutter up the user experience. Apple Maps now has a UI advantage (though not a data/search one, which matters more).
Apple is nickel-and-diming its customers. We spend thousands for their products and premium services like Apple One. We all know why: income must rise, and new firms are needed to scale. This will eventually backfire.

Jenn Leach
3 years ago
This clever Instagram marketing technique increased my sales to $30,000 per month.
No Paid Ads Required
I had an online store. After a year of running the company alongside my 9-to-5, I made enough to resign.
That day was amazing.
This Instagram marketing plan helped the store succeed.
How did I increase my sales to five figures a month without using any paid advertising?
I used customer event marketing.
I'm not sure this term exists. I invented it to describe what I was doing.
Instagram word-of-mouth, fan engagement, and interaction drove sales.
If a customer liked or disliked a product, the buzz would drive attention to the store.
I used customer-based events to increase engagement and store sales.
Success!
Here are the weekly Instagram customer events I coordinated while running my business:
Be the Buyer Days
Flash sales
Mystery boxes
Be the Buyer Days: How do they work?
Be the Buyer Days are exactly that.
You choose a day to share stock selections with social media followers.
This is an easy approach to engaging customers and getting fans enthusiastic about new releases.
First, pick a handful of items you’re considering ordering. I’d usually pick around 3 for Be the Buyer Day.
Then I'd poll the crowd on Instagram to vote on their favorites.
This was before Instagram stories, polls, and all the other cool features Instagram offers today. I think using these tools now would make this event even better.
I'd ask customers their favorite back then.
The growing comments excited customers.
Then I'd declare the winner, acquire the products, and start selling it.
How do flash sales work?
I mostly ran flash sales.
You choose a limited number of itemsdd for a few-hour sale.
We wanted most sales to result in sold-out items.
When an item sells out, it contributes to the sensation of scarcity and can inspire customers to visit your store to buy a comparable product, join your email list, become a fan, etc.
We hoped they'd act quickly.
I'd hold flash deals twice a week, which generated scarcity and boosted sales.
The store had a few thousand Instagram followers when I started flash deals.
Each flash sale item would make $400 to $600.
$400 x 3= $1,200
That's $1,200 on social media!
Twice a week, you'll make roughly $10K a month from Instagram.
$1,200/day x 8 events/month=$9,600
Flash sales did great.
We held weekly flash deals and sent social media and email reminders. That’s about it!
How are mystery boxes put together?
All you do is package a box of store products and sell it as a mystery box on TikTok or retail websites.
A $100 mystery box would cost $30.
You're discounting high-value boxes.
This is a clever approach to get rid of excess inventory and makes customers happy.
It worked!
Be the Buyer Days, flash deals, and mystery boxes helped build my company without paid advertisements.
All companies can use customer event marketing. Involving customers and providing an engaging environment can boost sales.
Try it!
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Atown Research
2 years ago
Meet the One-Person Businesses Earning Millions in Sales from Solo Founders
I've spent over 50 hours researching one-person firms, which interest me. I've found countless one-person enterprises that made millions on the founder's determination and perseverance.
Throughout my investigation, I found three of the most outstanding one-person enterprises. These enterprises show that people who work hard and dedicate themselves to their ideas may succeed.
Eric Barone (@ConcernedApe) created Stardew Valley in 2011 to better his job prospects. Eric loved making the game, in which players inherit a farm, grow crops, raise livestock, make friends with the villagers, and form a family.
Eric handled complete game production, including 3D graphics, animations, and music, to maintain creative control. He stopped job hunting and worked 8-15 hours a day on the game.
Eric developed a Stardew Valley website and subreddit to engage with gamers and get feedback. Eric's devoted community helped him meet Steam's minimum vote requirement for single creators.
Stardew Valley sold 1 million copies in two months after Eric launched it for $15 in 2016. The game has sold 20 million copies and made $300 million.
The game's inexpensive price, outsourcing of PR, marketing, and publication, and loyal player base helped it succeed. Eric has turned down million-dollar proposals from Sony and Nintendo to sell the game and instead updates and improves it. Haunted Chocolatier is Eric's new game.
Is farming not profitable? Ask Stardew Valley creator Eric Barone.
Gary Brewer established BuiltWith to assist users find website technologies and services. BuiltWith boasts 3000 paying customers and $14 million in yearly revenue, making it a significant resource for businesses wishing to generate leads, do customer analytics, obtain business insight, compare websites, or search websites by keyword.
BuiltWith has one full-time employee, Gary, and one or two part-time contractors that help with the blog. Gary handles sales, customer service, and other company functions alone.
BuiltWith acquired popularity through blog promotions and a top Digg ranking. About Us, a domain directory, connected to BuiltWith on every domain page, boosting it. Gary introduced $295–$995 monthly subscriptions to search technology, keywords, and potential consumers in response to customer demand.
Gary uses numerous methods to manage a firm without staff. He spends one to two hours every day answering user queries, most of which are handled quickly by linking to BuiltWiths knowledge store. Gary creates step-by-step essays or videos for complex problems. Gary can focus on providing new features based on customer comments and requests since he makes it easy to unsubscribe.
BuiltWith is entirely automated and successful due to its unique approach and useful offerings. It works for Google, Meta, Amazon, and Twitter.
Digital Inspiration develops Google Documents, Sheets, and Slides plugins. Digital Inspiration, founded by Amit Agarwal, receives 5 million monthly visits and earns $10 million. 40 million individuals have downloaded Digital Inspirations plugins.
Amit started Digital Inspiration by advertising his blog at tech events and getting Indian filter blogs and other newspapers to promote his articles. Amit built plugins and promoted them on the blog once the blog acquired popularity, using ideas from comments, friends, and Reddit. Digital Inspiration has over 20 free and premium plugins.
Mail Merge, Notifications for Google Forms, YouTube Uploader, and Document Studio are some of Digital Inspiration's most popular plugins. Mail Merge allows users to send personalized emails in bulk and track email opens and clicks.
Since Amits manages Digital Inspiration alone, his success is astounding. Amit developed a successful company via hard work and creativity, despite platform dependence. His tale inspires entrepreneurs.

Julie Plavnik
3 years ago
How to Become a Crypto Broker [Complying and Making Money]
Three options exist. The third one is the quickest and most fruitful.
You've mastered crypto trading and want to become a broker.
So you may wonder: Where to begin?
If so, keep reading.
Today I'll compare three different approaches to becoming a cryptocurrency trader.
What are cryptocurrency brokers, and how do they vary from stockbrokers?
A stockbroker implements clients' market orders (retail or institutional ones).
Brokerage firms are regulated, insured, and subject to regulatory monitoring.
Stockbrokers are required between buyers and sellers. They can't trade without a broker. To trade, a trader must open a broker account and deposit money. When a trader shops, he tells his broker what orders to place.
Crypto brokerage is trade intermediation with cryptocurrency.
In crypto trading, however, brokers are optional.
Crypto exchanges offer direct transactions. Open an exchange account (no broker needed) and make a deposit.
Question:
Since crypto allows DIY trading, why use a broker?
Let's compare cryptocurrency exchanges vs. brokers.
Broker versus cryptocurrency exchange
Most existing crypto exchanges are basically brokers.
Examine their primary services:
connecting purchasers and suppliers
having custody of clients' money (with the exception of decentralized cryptocurrency exchanges),
clearance of transactions.
Brokerage is comparable, don't you think?
There are exceptions. I mean a few large crypto exchanges that follow the stock exchange paradigm. They outsource brokerage, custody, and clearing operations. Classic exchange setups are rare in today's bitcoin industry.
Back to our favorite “standard” crypto exchanges. All-in-one exchanges and brokers. And usually, they operate under a broker or a broker-dealer license, save for the exchanges registered somewhere in a free-trade offshore paradise. Those don’t bother with any licensing.
What’s the sense of having two brokers at a time?
Better liquidity and trading convenience.
The crypto business is compartmentalized.
We have CEXs, DEXs, hybrid exchanges, and semi-exchanges (those that aggregate liquidity but do not execute orders on their sides). All have unique regulations and act as sovereign states.
There are about 18k coins and hundreds of blockchain protocols, most of which are heterogeneous (i.e., different in design and not interoperable).
A trader must register many accounts on different exchanges, deposit funds, and manage them all concurrently to access global crypto liquidity.
It’s extremely inconvenient.
Crypto liquidity fragmentation is the largest obstacle and bottleneck blocking crypto from mass adoption.
Crypto brokers help clients solve this challenge by providing one-gate access to deep and diverse crypto liquidity from numerous exchanges and suppliers. Professionals and institutions need it.
Another killer feature of a brokerage may be allowing clients to trade crypto with fiat funds exclusively, without fiat/crypto conversion. It is essential for professional and institutional traders.
Who may work as a cryptocurrency broker?
Apparently, not anyone. Brokerage requires high-powered specialists because it involves other people's money.
Here's the essentials:
excellent knowledge, skills, and years of trading experience
high-quality, quick, and secure infrastructure
highly developed team
outstanding trading capital
High-ROI network: long-standing, trustworthy connections with customers, exchanges, liquidity providers, payment gates, and similar entities
outstanding marketing and commercial development skills.
What about a license for a cryptocurrency broker? Is it necessary?
Complex question.
If you plan to play in white-glove jurisdictions, you may need a license. For example, in the US, as a “money transmitter” or as a CASSP (crypto asset secondary services provider) in Australia.
Even in these jurisdictions, there are no clear, holistic crypto brokerage and licensing policies.
Your lawyer will help you decide if your crypto brokerage needs a license.
Getting a license isn't quick. Two years of patience are needed.
How can you turn into a cryptocurrency broker?
Finally, we got there! 🎉
Three actionable ways exist:
To kickstart a regulated stand-alone crypto broker
To get a crypto broker franchise, and
To become a liquidity network broker.
Let's examine each.
1. Opening a regulated cryptocurrency broker
It's difficult. Especially If you're targeting first-world users.
You must comply with many regulatory, technical, financial, HR, and reporting obligations to keep your organization running. Some are mentioned above.
The licensing process depends on the products you want to offer (spots or derivatives) and the geographic areas you plan to service. There are no general rules for that.
In an overgeneralized way, here are the boxes you will have to check:
capital availability (usually a large amount of capital c is required)
You will have to move some of your team members to the nation providing the license in order to establish an office presence there.
the core team with the necessary professional training (especially applies to CEO, Head of Trading, Assistant to Head of Trading, etc.)
insurance
infrastructure that is trustworthy and secure
adopted proper AML/KYC/financial monitoring policies, etc.
Assuming you passed, what's next?
I bet it won’t be mind-blowing for you that the license is just a part of the deal. It won't attract clients or revenue.
To bring in high-dollar clientele, you must be a killer marketer and seller. It's not easy to convince people to give you money.
You'll need to be a great business developer to form successful, long-term agreements with exchanges (ideally for no fees), liquidity providers, banks, payment gates, etc. Persuade clients.
It's a tough job, isn't it?
I expect a Quora-type question here:
Can I start an unlicensed crypto broker?
Well, there is always a workaround with crypto!
You can register your broker in a free-trade zone like Seychelles to avoid US and other markets with strong watchdogs.
This is neither wise nor sustainable.
First, such experiments are illegal.
Second, you'll have trouble attracting clients and strategic partners.
A license equals trust. That’s it.
Even a pseudo-license from Mauritius matters.
Here are this method's benefits and downsides.
Cons first.
As you navigate this difficult and expensive legal process, you run the risk of missing out on business prospects. It's quite simple to become excellent compliance yet unable to work. Because your competitors are already courting potential customers while you are focusing all of your effort on paperwork.
Only God knows how long it will take you to pass the break-even point when everything with the license has been completed.
It is a money-burning business, especially in the beginning when the majority of your expenses will go toward marketing, sales, and maintaining license requirements. Make sure you have the fortitude and resources necessary to face such a difficult challenge.
Pros
It may eventually develop into a tool for making money. Because big guys who are professionals at trading require a white-glove regulated brokerage. You have every possibility if you work hard in the areas of sales, marketing, business development, and wealth. Simply put, everything must align.
Launching a regulated crypto broker is analogous to launching a crypto exchange. It's ROUGH. Sure you can take it?
2. Franchise for Crypto Broker (Crypto Sub-Brokerage)
A broker franchise is easier and faster than becoming a regulated crypto broker. Not a traditional brokerage.
A broker franchisee, often termed a sub-broker, joins with a broker (a franchisor) to bring them new clients. Sub-brokers market a broker's products and services to clients.
Sub-brokers are the middlemen between a broker and an investor.
Why is sub-brokering easier?
less demanding qualifications and legal complexity. All you need to do is keep a few certificates on hand (each time depends on the jurisdiction).
No significant investment is required
there is no demand that you be a trading member of an exchange, etc.
As a sub-broker, you can do identical duties without as many rights and certifications.
What about the crypto broker franchise?
Sub-brokers aren't common in crypto.
In most existing examples (PayBito, PCEX, etc.), franchises are offered by crypto exchanges, not brokers. Though we remember that crypto exchanges are, in fact, brokers, do we?
Similarly:
For a commission, a franchiser crypto broker receives new leads from a crypto sub-broker.
See above for why enrolling is easy.
Finding clients is difficult. Most crypto traders prefer to buy-sell on their own or through brokers over sub-broker franchises.
3. Broker of the Crypto Trading Network (or a Network Broker)
It's the greatest approach to execute crypto brokerage, based on effort/return.
Network broker isn't an established word. I wrote it for clarity.
Remember how we called crypto liquidity fragmentation the current crypto finance paradigm's main bottleneck?
Where there's a challenge, there's progress.
Several well-funded projects are aiming to fix crypto liquidity fragmentation. Instead of launching another crypto exchange with siloed trading, the greatest minds create trading networks that aggregate crypto liquidity from desynchronized sources and enable quick, safe, and affordable cross-blockchain transactions. Each project offers a distinct option for users.
Crypto liquidity implies:
One-account access to cryptocurrency liquidity pooled from network participants' exchanges and other liquidity sources
compiled price feeds
Cross-chain transactions that are quick and inexpensive, even for HFTs
link between participants of all kinds, and
interoperability among diverse blockchains
Fast, diversified, and cheap global crypto trading from one account.
How does a trading network help cryptocurrency brokers?
I’ll explain it, taking Yellow Network as an example.
Yellow provides decentralized Layer-3 peer-to-peer trading.
trade across chains globally with real-time settlement and
Between cryptocurrency exchanges, brokers, trading companies, and other sorts of network members, there is communication and the exchange of financial information.
Have you ever heard about ECN (electronic communication network)? If not, it's an automated system that automatically matches buy and sell orders. Yellow is a decentralized digital asset ECN.
Brokers can:
Start trading right now without having to meet stringent requirements; all you need to do is integrate with Yellow Protocol and successfully complete some KYC verification.
Access global aggregated crypto liquidity through a single point.
B2B (Broker to Broker) liquidity channels that provide peer liquidity from other brokers. Orders from the other broker will appear in the order book of a broker who is peering with another broker on the market. It will enable a broker to broaden his offer and raise the total amount of liquidity that is available to his clients.
Select a custodian or use non-custodial practices.
Comparing network crypto brokerage to other types:
A licensed stand-alone brokerage business is much more difficult and time-consuming to launch than network brokerage, and
Network brokerage, in contrast to crypto sub-brokerage, is scalable, independent, and offers limitless possibilities for revenue generation.
Yellow Network Whitepaper. has more details on how to start a brokerage business and what rewards you'll obtain.
Final thoughts
There are three ways to become a cryptocurrency broker, including the non-conventional liquidity network brokerage. The last option appears time/cost-effective.
Crypto brokerage isn't crowded yet. Act quickly to find your right place in this market.
Choose the way that works for you best and see you in crypto trading.
Discover Web3 & DeFi with Yellow Network!
Yellow, powered by Openware, is developing a cross-chain P2P liquidity aggregator to unite the crypto sector and provide global remittance services that aid people.
Join the Yellow Community and plunge into this decade's biggest product-oriented crypto project.
Observe Yellow Twitter
Enroll in Yellow Telegram
Visit Yellow Discord.
On Hacker Noon, look us up.
Yellow Network will expose development, technology, developer tools, crypto brokerage nodes software, and community liquidity mining.

Vitalik
4 years ago
An approximate introduction to how zk-SNARKs are possible (part 2)
If tasked with the problem of coming up with a zk-SNARK protocol, many people would make their way to this point and then get stuck and give up. How can a verifier possibly check every single piece of the computation, without looking at each piece of the computation individually? But it turns out that there is a clever solution.
Polynomials
Polynomials are a special class of algebraic expressions of the form:
- x+5
- x^4
- x^3+3x^2+3x+1
- 628x^{271}+318x^{270}+530x^{269}+…+69x+381
i.e. they are a sum of any (finite!) number of terms of the form cx^k
There are many things that are fascinating about polynomials. But here we are going to zoom in on a particular one: polynomials are a single mathematical object that can contain an unbounded amount of information (think of them as a list of integers and this is obvious). The fourth example above contained 816 digits of tau, and one can easily imagine a polynomial that contains far more.
Furthermore, a single equation between polynomials can represent an unbounded number of equations between numbers. For example, consider the equation A(x)+ B(x) = C(x). If this equation is true, then it's also true that:
- A(0)+B(0)=C(0)
- A(1)+B(1)=C(1)
- A(2)+B(2)=C(2)
- A(3)+B(3)=C(3)
And so on for every possible coordinate. You can even construct polynomials to deliberately represent sets of numbers so you can check many equations all at once. For example, suppose that you wanted to check:
- 12+1=13
- 10+8=18
- 15+8=23
- 15+13=28
You can use a procedure called Lagrange interpolation to construct polynomials A(x) that give (12,10,15,15) as outputs at some specific set of coordinates (eg. (0,1,2,3)), B(x) the outputs (1,8,8,13) on thos same coordinates, and so forth. In fact, here are the polynomials:
- A(x)=-2x^3+\frac{19}{2}x^2-\frac{19}{2}x+12
- B(x)=2x^3-\frac{19}{2}x^2+\frac{29}{2}x+1
- C(x)=5x+13
Checking the equation A(x)+B(x)=C(x) with these polynomials checks all four above equations at the same time.
Comparing a polynomial to itself
You can even check relationships between a large number of adjacent evaluations of the same polynomial using a simple polynomial equation. This is slightly more advanced. Suppose that you want to check that, for a given polynomial F, F(x+2)=F(x)+F(x+1) with the integer range {0,1…89} (so if you also check F(0)=F(1)=1, then F(100) would be the 100th Fibonacci number)
As polynomials, F(x+2)-F(x+1)-F(x) would not be exactly zero, as it could give arbitrary answers outside the range x={0,1…98}. But we can do something clever. In general, there is a rule that if a polynomial P is zero across some set S=\{x_1,x_2…x_n\} then it can be expressed as P(x)=Z(x)*H(x), where Z(x)=(x-x_1)*(x-x_2)*…*(x-x_n) and H(x) is also a polynomial. In other words, any polynomial that equals zero across some set is a (polynomial) multiple of the simplest (lowest-degree) polynomial that equals zero across that same set.
Why is this the case? It is a nice corollary of polynomial long division: the factor theorem. We know that, when dividing P(x) by Z(x), we will get a quotient Q(x) and a remainder R(x) is strictly less than that of Z(x). Since we know that P is zero on all of S, it means that R has to be zero on all of S as well. So we can simply compute R(x) via polynomial interpolation, since it's a polynomial of degree at most n-1 and we know n values (the zeros at S). Interpolating a polynomial with all zeroes gives the zero polynomial, thus R(x)=0 and H(x)=Q(x).
Going back to our example, if we have a polynomial F that encodes Fibonacci numbers (so F(x+2)=F(x)+F(x+1) across x=\{0,1…98\}), then I can convince you that F actually satisfies this condition by proving that the polynomial P(x)=F(x+2)-F(x+1)-F(x) is zero over that range, by giving you the quotient:
H(x)=\frac{F(x+2)-F(x+1)-F(x)}{Z(x)}
Where Z(x) = (x-0)*(x-1)*…*(x-98).
You can calculate Z(x) yourself (ideally you would have it precomputed), check the equation, and if the check passes then F(x) satisfies the condition!
Now, step back and notice what we did here. We converted a 100-step-long computation into a single equation with polynomials. Of course, proving the N'th Fibonacci number is not an especially useful task, especially since Fibonacci numbers have a closed form. But you can use exactly the same basic technique, just with some extra polynomials and some more complicated equations, to encode arbitrary computations with an arbitrarily large number of steps.
see part 3
