How I made $160,000 from non-fiction books
I've sold over 40,000 non-fiction books on Amazon and made over $160,000 in six years while writing on the side.
I have a full-time job and three young sons; I can't spend 40 hours a week writing. This article describes my journey.
I write mainly tech books:
Thanks to my readers, many wrote positive evaluations. Several are bestsellers.
A few have been adopted by universities as textbooks:
My books' passive income allows me more time with my family.
Knowing I could quit my job and write full time gave me more confidence. And I find purpose in my work (i am in christian ministry).
I'm always eager to write. When work is a dread or something bad happens, writing gives me energy. Writing isn't scary. In fact, I can’t stop myself from writing!
Writing has also established my tech authority. Universities use my books, as I've said. Traditional publishers have asked me to write books.
These mindsets helped me become a successful nonfiction author:
1. You don’t have to be an Authority
Yes, I have computer science experience. But I'm no expert on my topics. Before authoring "Beginning Node.js, Express & MongoDB," my most profitable book, I had no experience with those topics. Node was a new server-side technology for me. Would that stop me from writing a book? It can. I liked learning a new technology. So I read the top three Node books, took the top online courses, and put them into my own book (which makes me know more than 90 percent of people already).
I didn't have to worry about using too much jargon because I was learning as I wrote. An expert forgets a beginner's hardship.
"The fellow learner can aid more than the master since he knows less," says C.S. Lewis. The problem he must explain is recent. The expert has forgotten.”
2. Solve a micro-problem (Niching down)
I didn't set out to write a definitive handbook. I found a market with several challenges and wrote one book. Ex:
- Instead of web development, what about web development using Angular?
- Instead of Blockchain, what about Blockchain using Solidity and React?
- Instead of cooking recipes, how about a recipe for a specific kind of diet?
- Instead of Learning math, what about Learning Singapore Math?
3. Piggy Backing Trends
The above topics may still be a competitive market. E.g. Angular, React. To stand out, include the latest technologies or trends in your book. Learn iOS 15 instead of iOS programming. Instead of personal finance, what about personal finance with NFTs.
Even though you're a newbie author, your topic is well-known.
4. Publish short books
My books are known for being direct. Many people like this:
Your reader will appreciate you cutting out the fluff and getting to the good stuff. A reader can finish and review your book.
Second, short books are easier to write. Instead of creating a 500-page book for $50 (which few will buy), write a 100-page book that answers a subset of the problem and sell it for less. (You make less, but that's another subject). At least it got published instead of languishing. Less time spent creating a book means less time wasted if it fails. Write a small-bets book portfolio like Daniel Vassallo!
Third, it's $2.99-$9.99 on Amazon (gets 70 percent royalties for ebooks). Anything less receives 35% royalties. $9.99 books have 20,000–30,000 words. If you write more and charge more over $9.99, you get 35% royalties. Why not make it a $9.99 book?
(This is the ebook version.) Paperbacks cost more. Higher royalties allow for higher prices.
5. Validate book idea
Amazon will tell you if your book concept, title, and related phrases are popular. See? Check its best-sellers list.
150,000 is preferable. It sells 2–3 copies daily. Consider your rivals. Profitable niches have high demand and low competition.
Don't be afraid of competitive niches. First, it shows high demand. Secondly, what are the ways you can undercut the completion? Better book? Or cheaper option? There was lots of competition in my NodeJS book's area. None received 4.5 stars or more. I wrote a NodeJS book. Today, it's a best-selling Node book.
What’s Next
So long. Part II follows. Meanwhile, I will continue to write more books!
Follow my journey on Twitter.
This post is a summary. Read full article here
More on Entrepreneurship/Creators

Scrum Ventures
3 years ago
Trends from the Winter 2022 Demo Day at Y Combinators
Y Combinators Winter 2022 Demo Day continues the trend of more startups engaging in accelerator Demo Days. Our team evaluated almost 400 projects in Y Combinator's ninth year.
After Winter 2021 Demo Day, we noticed a hurry pushing shorter rounds, inflated valuations, and larger batches.
Despite the batch size, this event's behavior showed a return to normalcy. Our observations show that investors evaluate and fund businesses more carefully. Unlike previous years, more YC businesses gave investors with data rooms and thorough pitch decks in addition to valuation data before Demo Day.
Demo Day pitches were virtual and fast-paced, limiting unplanned meetings. Investors had more time and information to do their due research before meeting founders. Our staff has more time to study diverse areas and engage with interesting entrepreneurs and founders.
This was one of the most regionally diversified YC cohorts to date. This year's Winter Demo Day startups showed some interesting tendencies.
Trends and Industries to Watch Before Demo Day
Demo day events at any accelerator show how investment competition is influencing startups. As startups swiftly become scale-ups and big success stories in fintech, e-commerce, healthcare, and other competitive industries, entrepreneurs and early-stage investors feel pressure to scale quickly and turn a notion into actual innovation.
Too much eagerness can lead founders to focus on market growth and team experience instead of solid concepts, technical expertise, and market validation. Last year, YC Winter Demo Day funding cycles ended too quickly and valuations were unrealistically high.
Scrum Ventures observed a longer funding cycle this year compared to last year's Demo Day. While that seems promising, many factors could be contributing to change, including:
Market patterns are changing and the economy is becoming worse.
the industries that investors are thinking about.
Individual differences between each event batch and the particular businesses and entrepreneurs taking part
The Winter 2022 Batch's Trends
Each year, we also wish to examine trends among early-stage firms and YC event participants. More international startups than ever were anticipated to present at Demo Day.
Less than 50% of demo day startups were from the U.S. For the S21 batch, firms from outside the US were most likely in Latin America or Europe, however this year's batch saw a large surge in startups situated in Asia and Africa.
YC Startup Directory
163 out of 399 startups were B2B software and services companies. Financial, healthcare, and consumer startups were common.
Our team doesn't plan to attend every pitch or speak with every startup's founders or team members. Let's look at cleantech, Web3, and health and wellness startup trends.
Our Opinions Following Conversations with 87 Startups at Demo Day
In the lead-up to Demo Day, we spoke with 87 of the 125 startups going. Compared to B2C enterprises, B2B startups had higher average valuations. A few outliers with high valuations pushed B2B and B2C means above the YC-wide mean and median.
Many of these startups develop business and technology solutions we've previously covered. We've seen API, EdTech, creative platforms, and cybersecurity remain strong and increase each year.
While these persistent tendencies influenced the startups Scrum Ventures looked at and the founders we interacted with on Demo Day, new trends required more research and preparation. Let's examine cleantech, Web3, and health and wellness startups.
Hardware and software that is green
Cleantech enterprises demand varying amounts of funding for hardware and software. Although the same overarching trend is fueling the growth of firms in this category, each subgroup has its own strategy and technique for investigation and identifying successful investments.
Many cleantech startups we spoke to during the YC event are focused on helping industrial operations decrease or recycle carbon emissions.
Carbon Crusher: Creating carbon negative roads
Phase Biolabs: Turning carbon emissions into carbon negative products and carbon neutral e-fuels
Seabound: Capturing carbon dioxide emissions from ships
Fleetzero: Creating electric cargo ships
Impossible Mining: Sustainable seabed mining
Beyond Aero: Creating zero-emission private aircraft
Verdn: Helping businesses automatically embed environmental pledges for product and service offerings, boost customer engagement
AeonCharge: Allowing electric vehicle (EV) drivers to more easily locate and pay for EV charging stations
Phoenix Hydrogen: Offering a hydrogen marketplace and a connected hydrogen hub platform to connect supply and demand for hydrogen fuel and simplify hub planning and partner program expansion
Aklimate: Allowing businesses to measure and reduce their supply chain’s environmental impact
Pina Earth: Certifying and tracking the progress of businesses’ forestry projects
AirMyne: Developing machines that can reverse emissions by removing carbon dioxide from the air
Unravel Carbon: Software for enterprises to track and reduce their carbon emissions
Web3: NFTs, the metaverse, and cryptocurrency
Web3 technologies handle a wide range of business issues. This category includes companies employing blockchain technology to disrupt entertainment, finance, cybersecurity, and software development.
Many of these startups overlap with YC's FinTech trend. Despite this, B2C and B2B enterprises were evenly represented in Web3. We examined:
Stablegains: Offering consistent interest on cash balance from the decentralized finance (DeFi) market
LiquiFi: Simplifying token management with automated vesting contracts, tax reporting, and scheduling. For companies, investors, and finance & accounting
NFTScoring: An NFT trading platform
CypherD Wallet: A multichain wallet for crypto and NFTs with a non-custodial crypto debit card that instantly converts coins to USD
Remi Labs: Allowing businesses to more easily create NFT collections that serve as access to products, memberships, events, and more
Cashmere: A crypto wallet for Web3 startups to collaboratively manage funds
Chaingrep: An API that makes blockchain data human-readable and tokens searchable
Courtyard: A platform for securely storing physical assets and creating 3D representations as NFTs
Arda: “Banking as a Service for DeFi,” an API that FinTech companies can use to embed DeFi products into their platforms
earnJARVIS: A premium cryptocurrency management platform, allowing users to create long-term portfolios
Mysterious: Creating community-specific experiences for Web3 Discords
Winter: An embeddable widget that allows businesses to sell NFTs to users purchasing with a credit card or bank transaction
SimpleHash: An API for NFT data that provides compatibility across blockchains, standardized metadata, accurate transaction info, and simple integration
Lifecast: Tools that address motion sickness issues for 3D VR video
Gym Class: Virtual reality (VR) multiplayer basketball video game
WorldQL: An asset API that allows NFT creators to specify multiple in-game interpretations of their assets, increasing their value
Bonsai Desk: A software development kit (SDK) for 3D analytics
Campfire: Supporting virtual social experiences for remote teams
Unai: A virtual headset and Visual World experience
Vimmerse: Allowing creators to more easily create immersive 3D experiences
Fitness and health
Scrum Ventures encountered fewer health and wellness startup founders than Web3 and Cleantech. The types of challenges these organizations solve are still diverse. Several of these companies are part of a push toward customization in healthcare, an area of biotech set for growth for companies with strong portfolios and experienced leadership.
Here are several startups we considered:
Syrona Health: Personalized healthcare for women in the workplace
Anja Health: Personalized umbilical cord blood banking and stem cell preservation
Alfie: A weight loss program focused on men’s health that coordinates medical care, coaching, and “community-based competition” to help users lose an average of 15% body weight
Ankr Health: An artificial intelligence (AI)-enabled telehealth platform that provides personalized side effect education for cancer patients and data collection for their care teams
Koko — A personalized sleep program to improve at-home sleep analysis and training
Condition-specific telehealth platforms and programs:
Reviving Mind: Chronic care management covered by insurance and supporting holistic, community-oriented health care
Equipt Health: At-home delivery of prescription medical equipment to help manage chronic conditions like obstructive sleep apnea
LunaJoy: Holistic women’s healthcare management for mental health therapy, counseling, and medication
12 Startups from YC's Winter 2022 Demo Day to Watch
Bobidi: 10x faster AI model improvement
Artificial intelligence (AI) models have become a significant tool for firms to improve how well and rapidly they process data. Bobidi helps AI-reliant firms evaluate their models, boosting data insights in less time and reducing data analysis expenditures. The business has created a gamified community that offers a bug bounty for AI, incentivizing community members to test and find weaknesses in clients' AI models.
Magna: DeFi investment management and token vesting
Magna delivers rapid, secure token vesting so consumers may turn DeFi investments into primitives. Carta for Web3 allows enterprises to effortlessly distribute tokens to staff or investors. The Magna team hopes to allow corporations use locked tokens as collateral for loans, facilitate secondary liquidity so investors can sell shares on a public exchange, and power additional DeFi applications.
Perl Street: Funding for infrastructure
This Fintech firm intends to help hardware entrepreneurs get financing by [democratizing] structured finance, unleashing billions for sustainable infrastructure and next-generation hardware solutions. This network has helped hardware entrepreneurs achieve more than $140 million in finance, helping companies working on energy storage devices, EVs, and creating power infrastructure.
CypherD: Multichain cryptocurrency wallet
CypherD seeks to provide a multichain crypto wallet so general customers can explore Web3 products without knowledge hurdles. The startup's beta app lets consumers access crypto from EVM blockchains. The founders have crypto, financial, and startup experience.
Unravel Carbon: Enterprise carbon tracking and offsetting
Unravel Carbon's AI-powered decarbonization technology tracks companies' carbon emissions. Singapore-based startup focuses on Asia. The software can use any company's financial data to trace the supply chain and calculate carbon tracking, which is used to make regulatory disclosures and suggest carbon offsets.
LunaJoy: Precision mental health for women
LunaJoy helped women obtain mental health support throughout life. The platform combines data science to create a tailored experience, allowing women to access psychotherapy, medication management, genetic testing, and health coaching.
Posh: Automated EV battery recycling
Posh attempts to solve one of the EV industry's largest logistical difficulties. Millions of EV batteries will need to be decommissioned in the next decade, and their precious metals and residual capacity will go unused for some time. Posh offers automated, scalable lithium battery disassembly, making EV battery recycling more viable.
Unai: VR headset with 5x higher resolution
Unai stands apart from metaverse companies. Its VR headgear has five times the resolution of existing options and emphasizes human expression and interaction in a remote world. Maxim Perumal's method of latency reduction powers current VR headsets.
Palitronica: Physical infrastructure cybersecurity
Palitronica blends cutting-edge hardware and software to produce networked electronic systems that support crucial physical and supply chain infrastructure. The startup's objective is to build solutions that defend national security and key infrastructure from cybersecurity threats.
Reality Defender: Deepfake detection
Reality Defender alerts firms to bogus users and changed audio, video, and image files. Reality Deference's API and web app score material in real time to prevent fraud, improve content moderation, and detect deception.
Micro Meat: Infrastructure for the manufacture of cell-cultured meat
MicroMeat promotes sustainable meat production. The company has created technologies to scale up bioreactor-grown meat muscle tissue from animal cells. Their goal is to scale up cultured meat manufacturing so cultivated meat products can be brought to market feasibly and swiftly, boosting worldwide meat consumption.
Fleetzero: Electric cargo ships
This startup's battery technology will make cargo ships more sustainable and profitable. Fleetzero's electric cargo ships have five times larger profit margins than fossil fuel ships. Fleetzeros' founder has marine engineering, ship operations, and enterprise sales and business experience.

Simone Basso
3 years ago
How I set up my teams to be successful
After 10 years of working in scale-ups, I've embraced a few concepts for scaling Tech and Product teams.
First, cross-functionalize teams. Product Managers represent the business, Product Designers the consumer, and Engineers build.
I organize teams of 5-10 individuals, following AWS's two pizza teams guidelines, with a Product Trio guiding each.
If more individuals are needed to reach a goal, I group teams under a Product Trio.
With Engineering being the biggest group, Staff/Principal Engineers often support the Trio on cross-team technical decisions.
Product Managers, Engineering Managers, or Engineers in the team may manage projects (depending on the project or aim), but the trio is collectively responsible for the team's output and outcome.
Once the Product Trio model is created, roles, duties, team ceremonies, and cooperation models must be clarified.
Keep reporting lines by discipline. Line managers are accountable for each individual's advancement, thus it's crucial that they know the work in detail.
Cross-team collaboration becomes more important after 3 teams (15-30 people). Teams can easily diverge in how they write code, run ceremonies, and build products.
Establishing groups of people that are cross-team, but grouped by discipline and skills, sharing and agreeing on working practices becomes critical.
The “Spotify Guild” model has been where I’ve taken a lot of my inspiration from.
Last, establish a taxonomy for communication channels.
In Slack, I create one channel per team and one per guild (and one for me to have discussions with the team leads).
These are just some of the basic principles I follow to organize teams.
A book I particularly like about team types and how they interact with each other is https://teamtopologies.com/.

Jim Siwek
3 years ago
In 2022, can a lone developer be able to successfully establish a SaaS product?
In the early 2000s, I began developing SaaS. I helped launch an internet fax service that delivered faxes to email inboxes. Back then, it saved consumers money and made the procedure easier.
Google AdWords was young then. Anyone might establish a new website, spend a few hundred dollars on keywords, and see dozens of new paying clients every day. That's how we launched our new SaaS, and these clients stayed for years. Our early ROI was sky-high.
Changing times
The situation changed dramatically after 15 years. Our paid advertising cost $200-$300 for every new customer. Paid advertising takes three to four years to repay.
Fortunately, we still had tens of thousands of loyal clients. Good organic rankings gave us new business. We needed less sponsored traffic to run a profitable SaaS firm.
Is it still possible?
Since selling our internet fax firm, I've dreamed about starting a SaaS company. One I could construct as a lone developer and progressively grow a dedicated customer base, as I did before in a small team.
It seemed impossible to me. Solo startups couldn't afford paid advertising. SEO was tough. Even the worst SaaS startup ideas attracted VC funding. How could I compete with startups that could hire great talent and didn't need to make money for years (or ever)?
The One and Only Way to Learn
After years of talking myself out of SaaS startup ideas, I decided to develop and launch one. I needed to know if a solitary developer may create a SaaS app in 2022.
Thus, I did. I invented webwriter.ai, an AI-powered writing tool for website content, from hero section headlines to blog posts, this year. I soft-launched an MVP in July.
Considering the Issue
Now that I've developed my own fully capable SaaS app for site builders and developers, I wonder if it's still possible. Can webwriter.ai be successful?
I know webwriter.ai's proposal is viable because Jasper.ai and Grammarly are also AI-powered writing tools. With competition comes validation.
To Win, Differentiate
To compete with well-funded established brands, distinguish to stand out to a portion of the market. So I can speak directly to a target user, unlike larger competition.
I created webwriter.ai to help web builders and designers produce web content rapidly. This may be enough differentiation for now.
Budget-Friendly Promotion
When paid search isn't an option, we get inventive. There are more tools than ever to promote a new website.
Organic Results
on social media (Twitter, Instagram, TikTok, LinkedIn)
Marketing with content that is compelling
Link Creation
Listings in directories
references made in blog articles and on other websites
Forum entries
The Beginning of the Journey
As I've labored to construct my software, I've pondered a new mantra. Not sure where that originated from, but I like it. I'll live by it and teach my kids:
“Do the work.”
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Franz Schrepf
3 years ago
What I Wish I'd Known About Web3 Before Building
Cryptoland rollercoaster
I've lost money in crypto.
Unimportant.
The real issue: I didn’t understand how.
I'm surrounded with winners. To learn more, I created my own NFTs, currency, and DAO.
Web3 is a hilltop castle. Everything is valuable, decentralized, and on-chain.
The castle is Disneyland: beautiful in images, but chaotic with lengthy lines and kids spending too much money on dressed-up animals.
When the throng and businesses are gone, Disneyland still has enchantment.
The Real Story of Web3
NFTs
Scarcity. Scarce NFTs. That's their worth.
Skull. Rare-looking!
Nonsense.
Bored Ape Yacht Club vs. my NFTs?
Marketing.
BAYC is amazing, but not for the reasons people believe. Apecoin and Otherside's art, celebrity following, and innovation? Stunning.
No other endeavor captured the zeitgeist better. Yet how long did you think it took to actually mint the NFTs?
1 hour? Maybe a week for the website?
Minting NFTs is incredibly easy. Kid-friendly. Developers are rare. Think about that next time somebody posts “DevS dO SMt!?”
NFTs will remain popular. These projects are like our Van Goghs and Monets. Still, be wary. It still uses exclusivity and wash selling like the OG art market.
Not all NFTs are art-related.
Soulbound and anonymous NFTs could offer up new use cases. Property rights, privacy-focused ID, open-source project verification. Everything.
NFTs build online trust through ownership.
We just need to evolve from the apes first.
NFTs' superpower is marketing until then.
Crypto currency
What the hell is a token?
99% of people are clueless.
So I invested in both coins and tokens. Same same. Only that they are not.
Coins have their own blockchain and developer/validator community. It's hard.
Creating a token on top of a blockchain? Five minutes.
Most consumers don’t understand the difference, creating an arbitrage opportunity: pretend you’re a serious project without having developers on your payroll.
Few market sites help. Take a look. See any tokens?
There's a hint one click deeper.
Some tokens are legitimate. Some coins are bad investments.
Tokens are utilized for DAO governance and DApp payments. Still, know who's behind a token. They might be 12 years old.
Coins take time and money. The recent LUNA meltdown indicates that currency investing requires research.
DAOs
Decentralized Autonomous Organizations (DAOs) don't work as you assume.
Yes, members can vote.
A productive organization requires more.
I've observed two types of DAOs.
Total decentralization total dysfunction
Centralized just partially. Community-driven.
A core team executes the DAO's strategy and roadmap in successful DAOs. The community owns part of the organization, votes on decisions, and holds the team accountable.
DAOs are public companies.
Amazing.
A shareholder meeting's logistics are staggering. DAOs may hold anonymous, secure voting quickly. No need for intermediaries like banks to chase up every shareholder.
Successful DAOs aren't totally decentralized. Large-scale voting and collaboration have never been easier.
And that’s all that matters.
Scale, speed.
My Web3 learnings
Disneyland is enchanting. Web3 too.
In a few cycles, NFTs may be used to build trust, not clout. Not speculating with coins. DAOs run organizations, not themselves.
Finally, some final thoughts:
NFTs will be a very helpful tool for building trust online. NFTs are successful now because of excellent marketing.
Tokens are not the same as coins. Look into any project before making a purchase. Make sure it isn't run by three 9-year-olds piled on top of one another in a trench coat, at the very least.
Not entirely decentralized, DAOs. We shall see a future where community ownership becomes the rule rather than the exception once we acknowledge this fact.
Crypto Disneyland is a rollercoaster with loops that make you sick.
Always buckle up.
Have fun!

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
Scott Hickmann
3 years ago Draft
This is a draft
My wallpape
