More on Marketing

Tim Denning
3 years ago
I Posted Six Times a Day for 210 Days on Twitter. Here's What Happened.
I'd spend hours composing articles only to find out they were useless. Twitter solved the problem.
Twitter is wrinkled, say critics.
Nope. Writing is different. It won't make sense until you write there.
Twitter is resurgent. People are reading again. 15-second TikToks overloaded our senses.
After nuking my 20,000-follower Twitter account and starting again, I wrote every day for 210 days.
I'll explain.
I came across the strange world of microblogging.
Traditional web writing is filler-heavy.
On Twitter, you must be brief. I played Wordle.
Twitter Threads are the most popular writing format. Like a blog post. It reminds me of the famous broetry posts on LinkedIn a few years ago.
Threads combine tweets into an article.
Sharp, concise sentences
No regard for grammar
As important as the information is how the text looks.
Twitter Threads are like Michael Angelo's David monument. He chipped away at an enormous piece of marble until a man with a big willy appeared.
That's Twitter Threads.
I tried to remove unnecessary layers from several of my Wordpress blog posts. Then I realized something.
Tweeting from scratch is easier and more entertaining. It's quicker and makes you think more concisely.
Superpower: saying much with little words. My long-form writing has improved. My article sentences resemble tweets.
You never know what will happen.
Twitter's subcultures are odd. Best-performing tweets are strange.
Unusual trend: working alone and without telling anyone. It's a rebellion against Instagram influencers who share their every moment.
Early on, random thoughts worked:
My friend’s wife is Ukrainian. Her family are trapped in the warzone. He is devastated. And here I was complaining about my broken garage door. War puts everything in perspective. Today is a day to be grateful for peace.
Documenting what's happening triggers writing. It's not about viral tweets. Helping others matters.
There are numerous anonymous users.
Twitter uses pseudonyms.
You don't matter. On sites like LinkedIn, you must use your real name. Welcome to the Cyberpunk metaverse of Twitter :)
One daily piece of writing is a powerful habit.
Habits build creator careers. Read that again.
Twitter is an easy habit to pick up. If you can't tweet in one sentence, something's wrong. Easy-peasy-japanese.
Not what I tweeted, but my constancy, made the difference.
Daily writing is challenging, especially if your supervisor is on your back. Twitter encourages writing.
Tweets evolved as the foundation of all other material.
During my experiment, I enjoyed Twitter's speed.
Tweets get immediate responses, comments, and feedback. My popular tweets become newspaper headlines. I've also written essays from tweet discussions.
Sometimes the tweet and article were clear. Twitter sometimes helped me overcome writer's block.
I used to spend hours composing big things that had little real-world use.
Twitter helped me. No guessing. Data guides my coverage and validates concepts.
Test ideas on Twitter.
It took some time for my email list to grow.
Subscribers are a writer's lifeblood.
Without them, you're broke and homeless when Mark Zuckerberg tweaks the algorithms for ad dollars. Twitter has three ways to obtain email subscribers:
1. Add a link to your bio.
Twitter allows bio links (LinkedIn now does too). My eBook's landing page is linked. I collect emails there.
2. Start an online newsletter.
Twitter bought newsletter app Revue. They promote what they own.
I just established up a Revue email newsletter. I imported them weekly into my ConvertKit email list.
3. Create Twitter threads and include a link to your email list in the final tweet.
Write Twitter Threads and link the last tweet to your email list (example below).
Initial email subscribers were modest.
Numbers are growing. Twitter provides 25% of my new email subscribers. Some days, 50 people join.
Without them, my writing career is over. I'd be back at a 9-5 job begging for time off to spend with my newborn daughter. Nope.
Collect email addresses or die trying.
As insurance against unsubscribes and Zucks, use a second email list or Discord community.
What I still need to do
Twitter's fun. I'm wiser. I need to enable auto-replies and auto-DMs (direct messages).
This adds another way to attract subscribers. I schedule tweets with Tweet Hunter.
It’s best to go slow. People assume you're an internet marketer if you spam them with click requests.
A human internet marketer is preferable to a robot. My opinion.
210 days on Twitter taught me that. I plan to use the platform until I'm a grandfather unless Elon ruins it.

Sammy Abdullah
3 years ago
How to properly price SaaS
Price Intelligently put out amazing content on pricing your SaaS product. This blog's link to the whole report is worth reading. Our key takeaways are below.
Don't base prices on the competition. Competitor-based pricing has clear drawbacks. Their pricing approach is yours. Your company offers customers something unique. Otherwise, you wouldn't create it. This strategy is static, therefore you can't add value by raising prices without outpricing competitors. Look, but don't touch is the competitor-based moral. You want to know your competitors' prices so you're in the same ballpark, but they shouldn't guide your selections. Competitor-based pricing also drives down prices.
Value-based pricing wins. This is customer-based pricing. Value-based pricing looks outward, not inward or laterally at competitors. Your clients are the best source of pricing information. By valuing customer comments, you're focusing on buyers. They'll decide if your pricing and packaging are right. In addition to asking consumers about cost savings or revenue increases, look at data like number of users, usage per user, etc.
Value-based pricing increases prices. As you learn more about the client and your worth, you'll know when and how much to boost rates. Every 6 months, examine pricing.
Cloning top customers. You clone your consumers by learning as much as you can about them and then reaching out to comparable people or organizations. You can't accomplish this without knowing your customers. Segmenting and reproducing them requires as much detail as feasible. Offer pricing plans and feature packages for 4 personas. The top plan should state Contact Us. Your highest-value customers want more advice and support.
Question your 4 personas. What's the one item you can't live without? Which integrations matter most? Do you do analytics? Is support important or does your company self-solve? What's too cheap? What's too expensive?
Not everyone likes per-user pricing. SaaS organizations often default to per-user analytics. About 80% of companies utilizing per-user pricing should use an alternative value metric because their goods don't give more value with more users, so charging for them doesn't make sense.
At least 3:1 LTV/CAC. Break even on the customer within 2 years, and LTV to CAC is greater than 3:1. Because customer acquisition costs are paid upfront but SaaS revenues accrue over time, SaaS companies face an early financial shortfall while paying back the CAC.
ROI should be >20:1. Indeed. Ensure the customer's ROI is 20x the product's cost. Microsoft Office costs $80 a year, but consumers would pay much more to maintain it.
A/B Testing. A/B testing is guessing. When your pricing page varies based on assumptions, you'll upset customers. You don't have enough customers anyway. A/B testing optimizes landing pages, design decisions, and other site features when you know the problem but not pricing.
Don't discount. It cheapens the product, makes it permanent, and increases churn. By discounting, you're ruining your pricing analysis.

Joseph Mavericks
3 years ago
You Don't Have to Spend $250 on TikTok Ads Because I Did
900K impressions, 8K clicks, and $$$ orders…
I recently started dropshipping. Now that I own my business and can charge it as a business expense, it feels less like money wasted if it doesn't work. I also made t-shirts to sell. I intended to open a t-shirt store and had many designs on a hard drive. I read that Tiktok advertising had a high conversion rate and low cost because they were new. According to many, the advertising' cost/efficiency ratio would plummet and become as bad as Google or Facebook Ads. Now felt like the moment to try Tiktok marketing and dropshipping. I work in marketing for a SaaS firm and have seen how poorly ads perform. I wanted to try it alone.
I set up $250 and ran advertising for a week. Before that, I made my own products, store, and marketing. In this post, I'll show you my process and results.
Setting up the store
Dropshipping is a sort of retail business in which the manufacturer ships the product directly to the client through an online platform maintained by a seller. The seller takes orders but has no stock. The manufacturer handles all orders. This no-stock concept increases profitability and flexibility.
In my situation, I used previous t-shirt designs to make my own product. I didn't want to handle order fulfillment logistics, so I looked for a way to print my designs on demand, ship them, and handle order tracking/returns automatically. So I found Printful.
I needed to connect my backend and supplier to a storefront so visitors could buy. 99% of dropshippers use Shopify, but I didn't want to master the difficult application. I wanted a one-day project. I'd previously worked with Big Cartel, so I chose them.
Big Cartel doesn't collect commissions on sales, simply a monthly flat price ($9.99 to $19.99 depending on your plan).
After opening a Big Cartel account, I uploaded 21 designs and product shots, then synced each product with Printful.
Developing the ads
I mocked up my designs on cool people photographs from placeit.net, a great tool for creating product visuals when you don't have a studio, camera gear, or models to wear your t-shirts.
I opened an account on the website and had advertising visuals within 2 hours.
Because my designs are simple (black design on white t-shirt), I chose happy, stylish people on plain-colored backdrops. After that, I had to develop an animated slideshow.
Because I'm a graphic designer, I chose to use Adobe Premiere to create animated Tiktok advertising.
Premiere is a fancy video editing application used for more than advertisements. Premiere is used to edit movies, not social media marketing. I wanted this experiment to be quick, so I got 3 social media ad templates from motionarray.com and threw my visuals in. All the transitions and animations were pre-made in the files, so it only took a few hours to compile. The result:
I downloaded 3 different soundtracks for the videos to determine which would convert best.
After that, I opened a Tiktok business account, uploaded my films, and inserted ad info. They went live within one hour.
The (poor) outcomes
As a European company, I couldn't deliver ads in the US. All of my advertisements' material (title, description, and call to action) was in English, hence they continued getting rejected in Europe for countries that didn't speak English. There are a lot of them:
I lost a lot of quality traffic, but I felt that if the images were engaging, people would check out the store and buy my t-shirts. I was wrong.
51,071 impressions on Day 1. 0 orders after 411 clicks
114,053 impressions on Day 2. 1.004 clicks and no orders
Day 3: 987 clicks, 103,685 impressions, and 0 orders
101,437 impressions on Day 4. 0 orders after 963 clicks
115,053 impressions on Day 5. 1,050 clicks and no purchases
125,799 impressions on day 6. 1,184 clicks, no purchases
115,547 impressions on Day 7. 1,050 clicks and no purchases
121,456 impressions on day 8. 1,083 clicks, no purchases
47,586 impressions on Day 9. 419 Clicks. No orders
My overall conversion rate for video advertisements was 0.9%. TikTok's paid ad formats all result in strong engagement rates (ads average 3% to 12% CTR to site), therefore a 1 to 2% CTR should have been doable.
My one-week experiment yielded 8,151 ad clicks but no sales. Even if 0.1% of those clicks converted, I should have made 8 sales. Even companies with horrible web marketing would get one download or trial sign-up for every 8,151 clicks. I knew that because my advertising were in English, I had no impressions in the main EU markets (France, Spain, Italy, Germany), and that this impacted my conversion potential. I still couldn't believe my numbers.
I dug into the statistics and found that Tiktok's stats didn't match my store traffic data.
Looking more closely at the numbers
My ads were approved on April 26 but didn't appear until April 27. My store dashboard showed 440 visitors but 1,004 clicks on Tiktok. This happens often while tracking campaign results since different platforms handle comparable user activities (click, view) differently. In online marketing, residual data won't always match across tools.
My data gap was too large. Even if half of the 1,004 persons who clicked closed their browser or left before the store site loaded, I would have gained 502 visitors. The significant difference between Tiktok clicks and Big Cartel store visits made me suspicious. It happened all week:
Day 1: 440 store visits and 1004 ad clicks
Day 2: 482 store visits, 987 ad clicks
3rd day: 963 hits on ads, 452 store visits
443 store visits and 1,050 ad clicks on day 4.
Day 5: 459 store visits and 1,184 ad clicks
Day 6: 430 store visits and 1,050 ad clicks
Day 7: 409 store visits and 1,031 ad clicks
Day 8: 166 store visits and 418 ad clicks
The disparity wasn't related to residual data or data processing. The disparity between visits and clicks looked regular, but I couldn't explain it.
After the campaign concluded, I discovered all my creative assets (the videos) had a 0% CTR and a $0 expenditure in a separate dashboard. Whether it's a dashboard reporting issue or a budget allocation bug, online marketers shouldn't see this.
Tiktok can present any stats they want on their dashboard, just like any other platform that runs advertisements to promote content to its users. I can't verify that 895,687 individuals saw and clicked on my ad. I invested $200 for what appears to be around 900K impressions, which is an excellent ROI. No one bought a t-shirt, even an unattractive one, out of 900K people?
Would I do it again?
Nope. Whether I didn't make sales because Tiktok inflated the dashboard numbers or because I'm horrible at producing advertising and items that sell, I’ll stick to writing content and making videos. If setting up a business and ads in a few days was all it took to make money online, everyone would do it.
Video advertisements and dropshipping aren't dead. As long as the internet exists, people will click ads and buy stuff. Converting ads and selling stuff takes a lot of work, and I want to focus on other things.
I had always wanted to try dropshipping and I’m happy I did, I just won’t stick to it because that’s not something I’m interested in getting better at.
If I want to sell t-shirts again, I'll avoid Tiktok advertisements and find another route.
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Antonio Neto
3 years ago
What's up with tech?
Massive Layoffs, record low VC investment, debate over crash... why is it happening and what’s the endgame?
This article generalizes a diverse industry. For objectivity, specific tech company challenges like growing competition within named segments won't be considered. Please comment on the posts.
According to Layoffs.fyi, nearly 120.000 people have been fired from startups since March 2020. More than 700 startups have fired 1% to 100% of their workforce. "The tech market is crashing"
Venture capital investment dropped 19% QoQ in the first four months of 2022, a 2018 low. Since January 2022, Nasdaq has dropped 27%. Some believe the tech market is collapsing.
It's bad, but nothing has crashed yet. We're about to get super technical, so buckle up!
I've written a follow-up article about what's next. For a more optimistic view of the crisis' aftermath, see: Tech Diaspora and Silicon Valley crisis
What happened?
Insanity reigned. Last decade, everyone became a unicorn. Seed investments can be made without a product or team. While the "real world" economy suffered from the pandemic for three years, tech companies enjoyed the "new normal."
COVID sped up technology adoption on several fronts, but this "new normal" wasn't so new after many restrictions were lifted. Worse, it lived with disrupted logistics chains, high oil prices, and WW3. The consumer market has felt the industry's boom for almost 3 years. Inflation, unemployment, mental distress...what looked like a fast economic recovery now looks like unfulfilled promises.
People rethink everything they eat. Paying a Netflix subscription instead of buying beef is moronic if you can watch it for free on your cousin’s account. No matter how great your real estate app's UI is, buying a house can wait until mortgage rates drop. PLGProduct Led Growth (PLG) isn't the go-to strategy when consumers have more basic expense priorities.
Exponential growth and investment
Until recently, tech companies believed that non-exponential revenue growth was fatal. Exponential growth entails doing more with less. From Salim Ismail words:
An Exponential Organization (ExO) has 10x the impact of its peers.
Many tech companies' theories are far from reality.
Investors have funded (sometimes non-exponential) growth. Scale-driven companies throw people at problems until they're solved. Need an entire closing team because you’ve just bought a TV prime time add? Sure. Want gold-weight engineers to colorize buttons? Why not?
Tech companies don't need cash flow to do it; they can just show revenue growth and get funding. Even though it's hard to get funding, this was the market's momentum until recently.
The graph at the beginning of this section shows how industry heavyweights burned money until 2020, despite being far from their market-share seed stage. Being big and being sturdy are different things, and a lot of the tech startups out there are paper tigers. Without investor money, they have no foundation.
A little bit about interest rates
Inflation-driven high interest rates are said to be causing tough times. Investors would rather leave money in the bank than spend it (I myself said it some days ago). It’s not wrong, but it’s also not that simple.
The USA central bank (FED) is a good proxy of global economics. Dollar treasury bonds are the safest investment in the world. Buying U.S. debt, the only country that can print dollars, guarantees payment.
The graph above shows that FED interest rates are low and 10+ year bond yields are near 2018 levels. Nobody was firing at 2018. What’s with that then?
Full explanation is too technical for this article, so I'll just summarize: Bond yields rise due to lack of demand or market expectations of longer-lasting inflation. Safe assets aren't a "easy money" tactic for investors. If that were true, we'd have seen the current scenario before.
Long-term investors are protecting their capital from inflation.
Not a crash, a landing
I bombarded you with info... Let's review:
Consumption is down, hurting revenue.
Tech companies of all ages have been hiring to grow revenue at the expense of profit.
Investors expect inflation to last longer, reducing future investment gains.
Inflation puts pressure on a wheel that was rolling full speed not long ago. Investment spurs hiring, growth, and more investment. Worried investors and consumers reduce the cycle, and hiring follows.
Long-term investors back startups. When the invested company goes public or is sold, it's ok to burn money. What happens when the payoff gets further away? What if all that money sinks? Investors want immediate returns.
Why isn't the market crashing? Technology is not losing capital. It’s expecting change. The market realizes it threw moderation out the window and is reversing course. Profitability is back on the menu.
People solve problems and make money, but they also cost money. Huge cost for the tech industry. Engineers, Product Managers, and Designers earn up to 100% more than similar roles. Businesses must be careful about who they keep and in what positions to avoid wasting money.
What the future holds
From here on, it's all speculation. I found many great articles while researching this piece. Some are cited, others aren't (like this and this). We're in an adjustment period that may or may not last long.
Big companies aren't laying off many workers. Netflix firing 100 people makes headlines, but it's only 1% of their workforce. The biggest seem to prefer not hiring over firing.
Smaller startups beyond the seeding stage may be hardest hit. Without structure or product maturity, many will die.
I expect layoffs to continue for some time, even at Meta or Amazon. I don't see any industry names falling like they did during the .com crisis, but the market will shrink.
If you are currently employed, think twice before moving out and where to.
If you've been fired, hurry, there are still many opportunities.
If you're considering a tech career, wait.
If you're starting a business, I respect you. Good luck.

Jonathan Vanian
4 years ago
What is Terra? Your guide to the hot cryptocurrency
With cryptocurrencies like Bitcoin, Ether, and Dogecoin gyrating in value over the past few months, many people are looking at so-called stablecoins like Terra to invest in because of their more predictable prices.
Terraform Labs, which oversees the Terra cryptocurrency project, has benefited from its rising popularity. The company said recently that investors like Arrington Capital, Lightspeed Venture Partners, and Pantera Capital have pledged $150 million to help it incubate various crypto projects that are connected to Terra.
Terraform Labs and its partners have built apps that operate on the company’s blockchain technology that helps keep a permanent and shared record of the firm’s crypto-related financial transactions.
Here’s what you need to know about Terra and the company behind it.
What is Terra?
Terra is a blockchain project developed by Terraform Labs that powers the startup’s cryptocurrencies and financial apps. These cryptocurrencies include the Terra U.S. Dollar, or UST, that is pegged to the U.S. dollar through an algorithm.
Terra is a stablecoin that is intended to reduce the volatility endemic to cryptocurrencies like Bitcoin. Some stablecoins, like Tether, are pegged to more conventional currencies, like the U.S. dollar, through cash and cash equivalents as opposed to an algorithm and associated reserve token.
To mint new UST tokens, a percentage of another digital token and reserve asset, Luna, is “burned.” If the demand for UST rises with more people using the currency, more Luna will be automatically burned and diverted to a community pool. That balancing act is supposed to help stabilize the price, to a degree.
“Luna directly benefits from the economic growth of the Terra economy, and it suffers from contractions of the Terra coin,” Terraform Labs CEO Do Kwon said.
Each time someone buys something—like an ice cream—using UST, that transaction generates a fee, similar to a credit card transaction. That fee is then distributed to people who own Luna tokens, similar to a stock dividend.
Who leads Terra?
The South Korean firm Terraform Labs was founded in 2018 by Daniel Shin and Kwon, who is now the company’s CEO. Kwon is a 29-year-old former Microsoft employee; Shin now heads the Chai online payment service, a Terra partner. Kwon said many Koreans have used the Chai service to buy goods like movie tickets using Terra cryptocurrency.
Terraform Labs does not make money from transactions using its crypto and instead relies on outside funding to operate, Kwon said. It has raised $57 million in funding from investors like HashKey Digital Asset Group, Divergence Digital Currency Fund, and Huobi Capital, according to deal-tracking service PitchBook. The amount raised is in addition to the latest $150 million funding commitment announced on July 16.
What are Terra’s plans?
Terraform Labs plans to use Terra’s blockchain and its associated cryptocurrencies—including one pegged to the Korean won—to create a digital financial system independent of major banks and fintech-app makers. So far, its main source of growth has been in Korea, where people have bought goods at stores, like coffee, using the Chai payment app that’s built on Terra’s blockchain. Kwon said the company’s associated Mirror trading app is experiencing growth in China and Thailand.
Meanwhile, Kwon said Terraform Labs would use its latest $150 million in funding to invest in groups that build financial apps on Terra’s blockchain. He likened the scouting and investing in other groups as akin to a “Y Combinator demo day type of situation,” a reference to the popular startup pitch event organized by early-stage investor Y Combinator.
The combination of all these Terra-specific financial apps shows that Terraform Labs is “almost creating a kind of bank,” said Ryan Watkins, a senior research analyst at cryptocurrency consultancy Messari.
In addition to cryptocurrencies, Terraform Labs has a number of other projects including the Anchor app, a high-yield savings account for holders of the group’s digital coins. Meanwhile, people can use the firm’s associated Mirror app to create synthetic financial assets that mimic more conventional ones, like “tokenized” representations of corporate stocks. These synthetic assets are supposed to be helpful to people like “a small retail trader in Thailand” who can more easily buy shares and “get some exposure to the upside” of stocks that they otherwise wouldn’t have been able to obtain, Kwon said. But some critics have said the U.S. Securities and Exchange Commission may eventually crack down on synthetic stocks, which are currently unregulated.
What do critics say?
Terra still has a long way to go to catch up to bigger cryptocurrency projects like Ethereum.
Most financial transactions involving Terra-related cryptocurrencies have originated in Korea, where its founders are based. Although Terra is becoming more popular in Korea thanks to rising interest in its partner Chai, it’s too early to say whether Terra-related currencies will gain traction in other countries.
Terra’s blockchain runs on a “limited number of nodes,” said Messari’s Watkins, referring to the computers that help keep the system running. That helps reduce latency that may otherwise slow processing of financial transactions, he said.
But the tradeoff is that Terra is less “decentralized” than other blockchain platforms like Ethereum, which is powered by thousands of interconnected computing nodes worldwide. That could make Terra less appealing to some blockchain purists.

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
