More on Entrepreneurship/Creators

Davlin Knight
3 years ago
2 pitfalls to stay away from when launching a YouTube channel
You do not want to miss these
Stop! Stop it! Two things to avoid when starting a YouTube channel. Critical. Possible channel-killers Its future revenue.
I'll tell you now, so don't say "I wish I knew."
The Notorious Copyright Allegation
My YouTube channel received a copyright claim before I sold it. This claim was on a one-minute video I thought I'd changed enough to make mine, but the original owner disagreed.
It cost me thousands in ad revenue. Original owner got the profits.
Well, it wasn't your video, you say.
Touché.
I've learned. Sorta
I couldn't stop looking at the video's views. The video got 1,000,000 views without any revenue. I made 4 more similar videos.
If they didn't get copyrighted, I'd be rolling in dough.
You've spent a week editing and are uploading to YouTube. You're thrilled as you stand and stretch your back. You see the video just before publishing.
No way!
The red exclamation point on checks.
Copyright claim!
YouTube lets you publish, but you won't make money.
Sounds fair? Well, it is.
Copyright claims mean you stole someone's work. Song, image, or video clip.
We wouldn't want our content used for money.
The only problem with this is that almost everything belongs to someone else. I doubt some of the biggest creators are sitting down and making their music for their videos. That just seems really excessive when you could make a quick search on YouTube and download a song (I definitely don’t do this because that would be stealing).
So how do you defeat a copyright defense?
Even copyright-free songs on YouTube aren't guaranteed. Some copyrighted songs claim to be free.
Use YouTube's free music library or pay for a subscription to adobe stock, epidemic sound, or artlist.io.
Most of my videos have Nintendo music. Almost all game soundtracks are copyright-free and offer a variety of songs.
Restriction on age
Age restrictions are a must-avoid. A channel dies.
YouTube never suggests age-restricted videos.
Shadow banning means YouTube hides your content from subscribers and non-subscribers.
Keeping your channel family-friendly can help.
I hear you complaining that your channel isn't for kids. I agree. Not everyone has a clean mouth or creates content for minors.
YouTube has changed rapidly in recent years. Focusing on kids. Fewer big creators are using profanity or explicit content in videos. Not YouTube-worthy.
Youtube wants to be family-friendly. A family-friendly movie. It won't promote illegal content. Yes, it allows profanity.
Do I recommend avoiding no-no words in videos? Never. Okay. YouTube's policies are shaky. YouTube uses video content to determine ad suitability.
No joke. If you're serious about becoming a content creator, avoid profanity and inappropriate topics.
If your channel covers 18+ topics, like crime or commentary, censor as much as possible.
YouTube can be like walking on eggshells. You never know what is gonna upset the boss. So play it safe and try to avoid getting on their bad side.
Mr. Beast, Dream, Markplier, Faze Rug, and PewDewPie are popular creators. They maintain it family-friendly while entertaining fans.
You got this.

Desiree Peralta
3 years ago
Why Now Is Your Chance To Create A Millionaire Career
People don’t believe in influencers anymore; they need people like you.
Social media influencers have dominated for years. We've seen videos, images, and articles of *famous* individuals unwrapping, reviewing, and endorsing things.
This industry generates billions. This year, marketers spent $2.23 billion on Instagram, $1 million on Youtube, and $775 million on Tiktok. This marketing has helped start certain companies.
Influencers are dying, so ordinary people like us may take over this billion-dollar sector. Why?
Why influencers are perishing
Most influencers lie to their fans, especially on Instagram. Influencers' first purpose was to make their lives so flawless that others would want to buy their stuff.
In 2015, an Australian influencer with 600,000 followers went viral for revealing all her photos and everything she did to seem great before deleting her account.
“I dramatically edited the pictures, I manipulated the environements, and made my life look perfect in social media… I remember I obsessively checked the like count for a full week since uploading it, a selfie that now has close to 2,500 likes. It got 5 likes. This was when I was so hungry for social media validation … This was the reason why I quit social media: for me, personally, it consumed me. I wasn’t living in a 3D world.”
Influencers then lost credibility.
Influencers seem to live in a bubble, separate from us. Thanks to self-popularity love's and constant awareness campaigns, people find these people ridiculous.
Influencers are praised more for showing themselves as natural and common than for showing luxuries and lies.
Little by little, they are dying, making room for a new group to take advantage of this multi-million dollar business, which gives us (ordinary people) a big opportunity to grow on any content creation platform we want.
Why this is your chance to develop on any platform for creating content
In 2021, I wrote “Not everyone who talks about money is a Financial Advisor, be careful of who you take advice from,”. In it, I warned that not everyone with a large following is a reputable source of financial advice.
Other writers hated this post and said I was wrong.
People don't want Jeff Bezos or Elon Musk's counsel, they said. They prefer to hear about their neighbor's restroom problems or his closest friend's terrible business.
Real advice from regular folks.
And I found this was true when I returned to my independent YouTube channel and had more than 1000 followers after having abandoned it with fewer than 30 videos in 2021 since there were already many personal finance and travel channels and I thought mine wasn't special.
People appreciated my videos because I was a 20-something girl trying to make money online, and they believed my advice more than that of influencers with thousands of followers.
I think today is the greatest time to grow on any platform as an ordinary person. Normal individuals give honest recommendations about what works for them and look easier to make because they have the same options as us.
Nobody cares how a millionaire acquired a Lamborghini unless it's entertaining. Education works now. Real counsel from average people is replicable.
Many individuals don't appreciate how false influencers seem (unreal bodies and excessive surgery and retouching) since it makes them feel uneasy.
That's why body-positive advertisements have been so effective, but they've lost ground in places like Tiktok, where the audience wants more content from everyday people than influencers living amazing lives. More people will relate to your content if you appear genuine.
Last thoughts
Influencers are dwindling. People want more real people to give real advice and demonstrate an ordinary life.
People will enjoy anything you tell about your daily life as long as you provide value, and you can build a following rapidly if you're honest.
This is a millionaire industry that is getting more expensive and will go with what works, so stand out immediately.

DC Palter
2 years ago
How Will You Generate $100 Million in Revenue? The Startup Business Plan
A top-down company plan facilitates decision-making and impresses investors.
A startup business plan starts with the product, the target customers, how to reach them, and how to grow the business.
Bottom-up is terrific unless venture investors fund it.
If it can prove how it can exceed $100M in sales, investors will invest. If not, the business may be wonderful, but it's not venture capital-investable.
As a rule, venture investors only fund firms that expect to reach $100M within 5 years.
Investors get nothing until an acquisition or IPO. To make up for 90% of failed investments and still generate 20% annual returns, portfolio successes must exit with a 25x return. A $20M-valued company must be acquired for $500M or more.
This requires $100M in sales (or being on a nearly vertical trajectory to get there). The company has 5 years to attain that milestone and create the requisite ROI.
This motivates venture investors (venture funds and angel investors) to hunt for $100M firms within 5 years. When you pitch investors, you outline how you'll achieve that aim.
I'm wary of pitches after seeing a million hockey sticks predicting $5M to $100M in year 5 that never materialized. Doubtful.
Startups fail because they don't have enough clients, not because they don't produce a great product. That jump from $5M to $100M never happens. The company reaches $5M or $10M, growing at 10% or 20% per year. That's great, but not enough for a $500 million deal.
Once it becomes clear the company won’t reach orbit, investors write it off as a loss. When a corporation runs out of money, it's shut down or sold in a fire sale. The company can survive if expenses are trimmed to match revenues, but investors lose everything.
When I hear a pitch, I'm not looking for bright income projections but a viable plan to achieve them. Answer these questions in your pitch.
Is the market size sufficient to generate $100 million in revenue?
Will the initial beachhead market serve as a springboard to the larger market or as quicksand that hinders progress?
What marketing plan will bring in $100 million in revenue? Is the market diffuse and will cost millions of dollars in advertising, or is it one, focused market that can be tackled with a team of salespeople?
Will the business be able to bridge the gap from a small but fervent set of early adopters to a larger user base and avoid lock-in with their current solution?
Will the team be able to manage a $100 million company with hundreds of people, or will hypergrowth force the organization to collapse into chaos?
Once the company starts stealing market share from the industry giants, how will it deter copycats?
The requirement to reach $100M may be onerous, but it provides a context for difficult decisions: What should the product be? Where should we concentrate? who should we hire? Every strategic choice must consider how to reach $100M in 5 years.
Focusing on $100M streamlines investor pitches. Instead of explaining everything, focus on how you'll attain $100M.
As an investor, I know I'll lose my money if the startup doesn't reach this milestone, so the revenue prediction is the first thing I look at in a pitch deck.
Reaching the $100M goal needs to be the first thing the entrepreneur thinks about when putting together the business plan, the central story of the pitch, and the criteria for every important decision the company makes.
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Matthew Cluff
3 years ago
GTO Poker 101
"GTO" (Game Theory Optimal) has been used a lot in poker recently. To clarify its meaning and application, the aim of this article is to define what it is, when to use it when playing, what strategies to apply for how to play GTO poker, for beginner and more advanced players!
Poker GTO
In poker, you can choose between two main winning strategies:
Exploitative play maximizes expected value (EV) by countering opponents' sub-optimal plays and weaker tendencies. Yes, playing this way opens you up to being exploited, but the weaker opponents you're targeting won't change their game to counteract this, allowing you to reap maximum profits over the long run.
GTO (Game-Theory Optimal): You try to play perfect poker, which forces your opponents to make mistakes (which is where almost all of your profit will be derived from). It mixes bluffs or semi-bluffs with value bets, clarifies bet sizes, and more.
GTO vs. Exploitative: Which is Better in Poker?
Before diving into GTO poker strategy, it's important to know which of these two play styles is more profitable for beginners and advanced players. The simple answer is probably both, but usually more exploitable.
Most players don't play GTO poker and can be exploited in their gameplay and strategy, allowing for more profits to be made using an exploitative approach. In fact, it’s only in some of the largest games at the highest stakes that GTO concepts are fully utilized and seen in practice, and even then, exploitative plays are still sometimes used.
Knowing, understanding, and applying GTO poker basics will create a solid foundation for your poker game. It's also important to understand GTO so you can deviate from it to maximize profits.
GTO Poker Strategy
According to Ed Miller's book "Poker's 1%," the most fundamental concept that only elite poker players understand is frequency, which could be in relation to cbets, bluffs, folds, calls, raises, etc.
GTO poker solvers (downloadable online software) give solutions for how to play optimally in any given spot and often recommend using mixed strategies based on select frequencies.
In a river situation, a solver may tell you to call 70% of the time and fold 30%. It may also suggest calling 50% of the time, folding 35% of the time, and raising 15% of the time (with a certain range of hands).
Frequencies are a fundamental and often unrecognized part of poker, but they run through these 5 GTO concepts.
1. Preflop ranges
To compensate for positional disadvantage, out-of-position players must open tighter hand ranges.
Premium starting hands aren't enough, though. Considering GTO poker ranges and principles, you want a good, balanced starting hand range from each position with at least some hands that can make a strong poker hand regardless of the flop texture (low, mid, high, disconnected, etc).
Below is a GTO preflop beginner poker chart for online 6-max play, showing which hand ranges one should open-raise with. Table positions are color-coded (see key below).
NOTE: For GTO play, it's advisable to use a mixed strategy for opening in the small blind, combining open-limps and open-raises for various hands. This cannot be illustrated with the color system used for the chart.
Choosing which hands to play is often a math problem, as discussed below.
Other preflop GTO poker charts include which hands to play after a raise, which to 3bet, etc. Solvers can help you decide which preflop hands to play (call, raise, re-raise, etc.).
2. Pot Odds
Always make +EV decisions that profit you as a poker player. Understanding pot odds (and equity) can help.
Postflop Pot Odds
Let’s say that we have JhTh on a board of 9h8h2s4c (open-ended straight-flush draw). We have $40 left and $50 in the pot. He has you covered and goes all-in. As calling or folding are our only options, playing GTO involves calculating whether a call is +EV or –EV. (The hand was empty.)
Any remaining heart, Queen, or 7 wins the hand. This means we can improve 15 of 46 unknown cards, or 32.6% of the time.
What if our opponent has a set? The 4h or 2h could give us a flush, but it could also give the villain a boat. If we reduce outs from 15 to 14.5, our equity would be 31.5%.
We must now calculate pot odds.
(bet/(our bet+pot)) = pot odds
= $50 / ($40 + $90)
= $40 / $130
= 30.7%
To make a profitable call, we need at least 30.7% equity. This is a profitable call as we have 31.5% equity (even if villain has a set). Yes, we will lose most of the time, but we will make a small profit in the long run, making a call correct.
Pot odds aren't just for draws, either. If an opponent bets 50% pot, you get 3 to 1 odds on a call, so you must win 25% of the time to be profitable. If your current hand has more than 25% equity against your opponent's perceived range, call.
Preflop Pot Odds
Preflop, you raise to 3bb and the button 3bets to 9bb. You must decide how to act. In situations like these, we can actually use pot odds to assist our decision-making.
This pot is:
(our open+3bet size+small blind+big blind)
(3bb+9bb+0.5bb+1bb)
= 13.5
This means we must call 6bb to win a pot of 13.5bb, which requires 30.7% equity against the 3bettor's range.
Three additional factors must be considered:
Being out of position on our opponent makes it harder to realize our hand's equity, as he can use his position to put us in tough spots. To profitably continue against villain's hand range, we should add 7% to our equity.
Implied Odds / Reverse Implied Odds: The ability to win or lose significantly more post-flop (than pre-flop) based on our remaining stack.
While statistics on 3bet stats can be gained with a large enough sample size (i.e. 8% 3bet stat from button), the numbers don't tell us which 8% of hands villain could be 3betting with. Both polarized and depolarized charts below show 8% of possible hands.
7.4% of hands are depolarized.
Polarized Hand range (7.54%):
Each hand range has different contents. We don't know if he 3bets some hands and calls or folds others.
Using an exploitable strategy can help you play a hand range correctly. The next GTO concept will make things easier.
3. Minimum Defense Frequency:
This concept refers to the % of our range we must continue with (by calling or raising) to avoid being exploited by our opponents. This concept is most often used off-table and is difficult to apply in-game.
These beginner GTO concepts will help your decision-making during a hand, especially against aggressive opponents.
MDF formula:
MDF = POT SIZE/(POT SIZE+BET SIZE)
Here's a poker GTO chart of common bet sizes and minimum defense frequency.
Take the number of hand combos in your starting hand range and use the MDF to determine which hands to continue with. Choose hands with the most playability and equity against your opponent's betting range.
Say you open-raise HJ and BB calls. Qh9h6c flop. Your opponent leads you for a half-pot bet. MDF suggests keeping 67% of our range.
Using the above starting hand chart, we can determine that the HJ opens 254 combos:
We must defend 67% of these hands, or 170 combos, according to MDF. Hands we should keep include:
Flush draws
Open-Ended Straight Draws
Gut-Shot Straight Draws
Overcards
Any Pair or better
So, our flop continuing range could be:
Some highlights:
Fours and fives have little chance of improving on the turn or river.
We only continue with AX hearts (with a flush draw) without a pair or better.
We'll also include 4 AJo combos, all of which have the Ace of hearts, and AcJh, which can block a backdoor nut flush combo.
Let's assume all these hands are called and the turn is blank (2 of spades). Opponent bets full-pot. MDF says we must defend 50% of our flop continuing range, or 85 of 170 combos, to be unexploitable. This strategy includes our best flush draws, straight draws, and made hands.
Here, we keep combining:
Nut flush draws
Pair + flush draws
GS + flush draws
Second Pair, Top Kicker+
One combo of JJ that doesn’t block the flush draw or backdoor flush draw.
On the river, we can fold our missed draws and keep our best made hands. When calling with weaker hands, consider blocker effects and card removal to avoid overcalling and decide which combos to continue.
4. Poker GTO Bet Sizing
To avoid being exploited, balance your bluffs and value bets. Your betting range depends on how much you bet (in relation to the pot). This concept only applies on the river, as draws (bluffs) on the flop and turn still have equity (and are therefore total bluffs).
On the flop, you want a 2:1 bluff-to-value-bet ratio. On the flop, there won't be as many made hands as on the river, and your bluffs will usually contain equity. The turn should have a "bluffing" ratio of 1:1. Use the chart below to determine GTO river bluff frequencies (relative to your bet size):
This chart relates to your opponent's pot odds. If you bet 50% pot, your opponent gets 3:1 odds and must win 25% of the time to call. Poker GTO theory suggests including 25% bluff combinations in your betting range so you're indifferent to your opponent calling or folding.
Best river bluffs don't block hands you want your opponent to have (or not have). For example, betting with missed Ace-high flush draws is often a mistake because you block a missed flush draw you want your opponent to have when bluffing on the river (meaning that it would subsequently be less likely he would have it, if you held two of the flush draw cards). Ace-high usually has some river showdown value.
If you had a 3-flush on the river and wanted to raise, you could bluff raise with AX combos holding the bluff suit Ace. Blocking the nut flush prevents your opponent from using that combo.
5. Bet Sizes and Frequency
GTO beginner strategies aren't just bluffs and value bets. They show how often and how much to bet in certain spots. Top players have benefited greatly from poker solvers, which we'll discuss next.
GTO Poker Software
In recent years, various poker GTO solvers have been released to help beginner, intermediate, and advanced players play balanced/GTO poker in various situations.
PokerSnowie and PioSolver are popular GTO and poker study programs.
While you can't compute players' hand ranges and what hands to bet or check with in real time, studying GTO play strategies with these programs will pay off. It will improve your poker thinking and understanding.
Solvers can help you balance ranges, choose optimal bet sizes, and master cbet frequencies.
GTO Poker Tournament
Late-stage tournaments have shorter stacks than cash games. In order to follow GTO poker guidelines, Nash charts have been created, tweaked, and used for many years (and also when to call, depending on what number of big blinds you have when you find yourself shortstacked).
The charts are for heads-up push/fold. In a multi-player game, the "pusher" chart can only be used if play is folded to you in the small blind. The "caller" chart can only be used if you're in the big blind and assumes a small blind "pusher" (with a much wider range than if a player in another position was open-shoving).
Divide the pusher chart's numbers by 2 to see which hand to use from the Button. Divide the original chart numbers by 4 to find the CO's pushing range. Some of the figures will be impossible to calculate accurately for the CO or positions to the right of the blinds because the chart's highest figure is "20+" big blinds, which is also used for a wide range of hands in the push chart.
Both of the GTO charts below are ideal for heads-up play, but exploitable HU shortstack strategies can lead to more +EV decisions against certain opponents. Following the charts will make your play GTO and unexploitable.
Poker pro Max Silver created the GTO push/fold software SnapShove. (It's accessible online at www.snapshove.com or as iOS or Android apps.)
Players can access GTO shove range examples in the full version. (You can customize the number of big blinds you have, your position, the size of the ante, and many other options.)
In Conclusion
Due to the constantly changing poker landscape, players are always improving their skills. Exploitable strategies often yield higher profit margins than GTO-based approaches, but knowing GTO beginner and advanced concepts can give you an edge for a few reasons.
It creates a solid gameplay base.
Having a baseline makes it easier to exploit certain villains.
You can avoid leveling wars with your opponents by making sound poker decisions based on GTO strategy.
It doesn't require assuming opponents' play styles.
Not results-oriented.
This is just the beginning of GTO and poker theory. Consider investing in the GTO poker solver software listed above to improve your game.
Matt Nutsch
3 years ago
Most people are unaware of how artificial intelligence (A.I.) is changing the world.
Recently, I saw an interesting social media post. In an entrepreneurship forum. A blogger asked for help because he/she couldn't find customers. I now suspect that the writer’s occupation is being disrupted by A.I.
Introduction
Artificial Intelligence (A.I.) has been a hot topic since the 1950s. With recent advances in machine learning, A.I. will touch almost every aspect of our lives. This article will discuss A.I. technology and its social and economic implications.
What's AI?
A computer program or machine with A.I. can think and learn. In general, it's a way to make a computer smart. Able to understand and execute complex tasks. Machine learning, NLP, and robotics are common types of A.I.
AI's global impact
AI will change the world, but probably faster than you think. A.I. already affects our daily lives. It improves our decision-making, efficiency, and productivity.
A.I. is transforming our lives and the global economy. It will create new business and job opportunities but eliminate others. Affected workers may face financial hardship.
AI examples:
OpenAI's GPT-3 text-generation
Developers can train, deploy, and manage models on GPT-3. It handles data preparation, model training, deployment, and inference for machine learning workloads. GPT-3 is easy to use for both experienced and new data scientists.
My team conducted an experiment. We needed to generate some blog posts for a website. We hired a blogger on Upwork. OpenAI created a blog post. The A.I.-generated blog post was of higher quality and lower cost.
MidjourneyAI's Art Contests
AI already affects artists. Artists use A.I. to create realistic 3D images and videos for digital art. A.I. is also used to generate new art ideas and methods.
MidjourneyAI and GigapixelAI won a contest last month. It's AI. created a beautiful piece of art that captured the contest's spirit. AI triumphs. It could open future doors.
After the art contest win, I registered to try out these new image generating A.I.s. In the MidjourneyAI chat forum, I noticed an artist's plea. The artist begged others to stop flooding RedBubble with AI-generated art.
Shutterstock and Getty Images have halted user uploads. AI-generated images flooded online marketplaces.
Imagining Videos with Meta
Meta released Make-a-Video this week. It's an A.I. app that creates videos from text. What you type creates a video.
This technology will impact TV, movies, and video games greatly. Imagine a movie or game that's personalized to your tastes. It's closer than you think.
Uses and Abuses of Deepfakes
Deepfake videos are computer-generated images of people. AI creates realistic images and videos of people.
Deepfakes are entertaining but have social implications. Porn introduced deepfakes in 2017. People put famous faces on porn actors and actresses without permission.
Soon, deepfakes were used to show dead actors/actresses or make them look younger. Carrie Fischer was included in films after her death using deepfake technology.
Deepfakes can be used to create fake news or manipulate public opinion, according to an AI.
Voices for Darth Vader and Iceman
James Earl Jones, who voiced Darth Vader, sold his voice rights this week. Aged actor won't be in those movies. Respeecher will use AI to mimic Jones's voice. This technology could change the entertainment industry. One actor can now voice many characters.
AI can generate realistic voice audio from text. Top Gun 2 actor Val Kilmer can't speak for medical reasons. Sonantic created Kilmer's voice from the movie script. This entertaining technology has social implications. It blurs authentic recordings and fake media.
Medical A.I. fights viruses
A team of Chinese scientists used machine learning to predict effective antiviral drugs last year. They started with a large dataset of virus-drug interactions. Researchers combined that with medication and virus information. Finally, they used machine learning to predict effective anti-virus medicines. This technology could solve medical problems.
AI ideas AI-generated Itself
OpenAI's GPT-3 predicted future A.I. uses. Here's what it told me:
AI will affect the economy. Businesses can operate more efficiently and reinvest resources with A.I.-enabled automation. AI can automate customer service tasks, reducing costs and improving satisfaction.
A.I. makes better pricing, inventory, and marketing decisions. AI automates tasks and makes decisions. A.I.-powered robots could help the elderly or disabled. Self-driving cars could reduce accidents.
A.I. predictive analytics can predict stock market or consumer behavior trends and patterns. A.I. also personalizes recommendations. sways. A.I. recommends products and movies. AI can generate new ideas based on data analysis.
Conclusion
A.I. will change business as it becomes more common. It will change how we live and work by creating growth and prosperity.
Exciting times, but also one which should give us all pause. Technology can be good or evil. We must use new technologies ethically, fairly, and honestly.
“The author generated some sentences in this text in part with GPT-3, OpenAI’s large-scale language-generation model. Upon generating draft language, the author reviewed, edited, and revised the language to their own liking and takes ultimate responsibility for the content of this publication. The text of this post was further edited using HemingWayApp. Many of the images used were generated using A.I. as described in the captions.”

Vitalik
3 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
