More on Economics & Investing

Theresa W. Carey
3 years ago
How Payment for Order Flow (PFOF) Works
What is PFOF?
PFOF is a brokerage firm's compensation for directing orders to different parties for trade execution. The brokerage firm receives fractions of a penny per share for directing the order to a market maker.
Each optionable stock could have thousands of contracts, so market makers dominate options trades. Order flow payments average less than $0.50 per option contract.
Order Flow Payments (PFOF) Explained
The proliferation of exchanges and electronic communication networks has complicated equity and options trading (ECNs) Ironically, Bernard Madoff, the Ponzi schemer, pioneered pay-for-order-flow.
In a December 2000 study on PFOF, the SEC said, "Payment for order flow is a method of transferring trading profits from market making to brokers who route customer orders to specialists for execution."
Given the complexity of trading thousands of stocks on multiple exchanges, market making has grown. Market makers are large firms that specialize in a set of stocks and options, maintaining an inventory of shares and contracts for buyers and sellers. Market makers are paid the bid-ask spread. Spreads have narrowed since 2001, when exchanges switched to decimals. A market maker's ability to play both sides of trades is key to profitability.
Benefits, requirements
A broker receives fees from a third party for order flow, sometimes without a client's knowledge. This invites conflicts of interest and criticism. Regulation NMS from 2005 requires brokers to disclose their policies and financial relationships with market makers.
Your broker must tell you if it's paid to send your orders to specific parties. This must be done at account opening and annually. The firm must disclose whether it participates in payment-for-order-flow and, upon request, every paid order. Brokerage clients can request payment data on specific transactions, but the response takes weeks.
Order flow payments save money. Smaller brokerage firms can benefit from routing orders through market makers and getting paid. This allows brokerage firms to send their orders to another firm to be executed with other orders, reducing costs. The market maker or exchange benefits from additional share volume, so it pays brokerage firms to direct traffic.
Retail investors, who lack bargaining power, may benefit from order-filling competition. Arrangements to steer the business in one direction invite wrongdoing, which can erode investor confidence in financial markets and their players.
Pay-for-order-flow criticism
It has always been controversial. Several firms offering zero-commission trades in the late 1990s routed orders to untrustworthy market makers. During the end of fractional pricing, the smallest stock spread was $0.125. Options spreads widened. Traders found that some of their "free" trades cost them a lot because they weren't getting the best price.
The SEC then studied the issue, focusing on options trades, and nearly decided to ban PFOF. The proliferation of options exchanges narrowed spreads because there was more competition for executing orders. Options market makers said their services provided liquidity. In its conclusion, the report said, "While increased multiple-listing produced immediate economic benefits to investors in the form of narrower quotes and effective spreads, these improvements have been muted with the spread of payment for order flow and internalization."
The SEC allowed payment for order flow to continue to prevent exchanges from gaining monopoly power. What would happen to trades if the practice was outlawed was also unclear. SEC requires brokers to disclose financial arrangements with market makers. Since then, the SEC has watched closely.
2020 Order Flow Payment
Rule 605 and Rule 606 show execution quality and order flow payment statistics on a broker's website. Despite being required by the SEC, these reports can be hard to find. The SEC mandated these reports in 2005, but the format and reporting requirements have changed over the years, most recently in 2018.
Brokers and market makers formed a working group with the Financial Information Forum (FIF) to standardize order execution quality reporting. Only one retail brokerage (Fidelity) and one market maker remain (Two Sigma Securities). FIF notes that the 605/606 reports "do not provide the level of information that allows a retail investor to gauge how well a broker-dealer fills a retail order compared to the NBBO (national best bid or offer’) at the time the order was received by the executing broker-dealer."
In the first quarter of 2020, Rule 606 reporting changed to require brokers to report net payments from market makers for S&P 500 and non-S&P 500 equity trades and options trades. Brokers must disclose payment rates per 100 shares by order type (market orders, marketable limit orders, non-marketable limit orders, and other orders).
Richard Repetto, Managing Director of New York-based Piper Sandler & Co., publishes a report on Rule 606 broker reports. Repetto focused on Charles Schwab, TD Ameritrade, E-TRADE, and Robinhood in Q2 2020. Repetto reported that payment for order flow was higher in the second quarter than the first due to increased trading activity, and that options paid more than equities.
Repetto says PFOF contributions rose overall. Schwab has the lowest options rates, while TD Ameritrade and Robinhood have the highest. Robinhood had the highest equity rating. Repetto assumes Robinhood's ability to charge higher PFOF reflects their order flow profitability and that they receive a fixed rate per spread (vs. a fixed rate per share by the other brokers).
Robinhood's PFOF in equities and options grew the most quarter-over-quarter of the four brokers Piper Sandler analyzed, as did their implied volumes. All four brokers saw higher PFOF rates.
TD Ameritrade took the biggest income hit when cutting trading commissions in fall 2019, and this report shows they're trying to make up the shortfall by routing orders for additional PFOF. Robinhood refuses to disclose trading statistics using the same metrics as the rest of the industry, offering only a vague explanation on their website.
Summary
Payment for order flow has become a major source of revenue as brokers offer no-commission equity (stock and ETF) orders. For retail investors, payment for order flow poses a problem because the brokerage may route orders to a market maker for its own benefit, not the investor's.
Infrequent or small-volume traders may not notice their broker's PFOF practices. Frequent traders and those who trade larger quantities should learn about their broker's order routing system to ensure they're not losing out on price improvement due to a broker prioritizing payment for order flow.
This post is a summary. Read full article here

Cory Doctorow
3 years ago
The current inflation is unique.
New Stiglitz just dropped.
Here's the inflation story everyone believes (warning: it's false): America gave the poor too much money during the recession, and now the economy is awash with free money, which made them so rich they're refusing to work, meaning the economy isn't making anything. Prices are soaring due to increased cash and missing labor.
Lawrence Summers says there's only one answer. We must impoverish the poor: raise interest rates, cause a recession, and eliminate millions of jobs, until the poor are stripped of their underserved fortunes and return to work.
https://pluralistic.net/2021/11/20/quiet-part-out-loud/#profiteering
This is nonsense. Countries around the world suffered inflation during and after lockdowns, whether they gave out humanitarian money to keep people from starvation. America has slightly greater inflation than other OECD countries, but it's not due to big relief packages.
The Causes of and Responses to Today's Inflation, a Roosevelt Institute report by Nobel-winning economist Joseph Stiglitz and macroeconomist Regmi Ira, debunks this bogus inflation story and offers a more credible explanation for inflation.
https://rooseveltinstitute.org/wp-content/uploads/2022/12/RI CausesofandResponsestoTodaysInflation Report 202212.pdf
Sharp interest rate hikes exacerbate the slump and increase inflation, the authors argue. They compare monetary policy inflation cures to medieval bloodletting, where doctors repeated the same treatment until the patient recovered (for which they received credit) or died (which was more likely).
Let's discuss bloodletting. Inflation hawks warn of the wage price spiral, when inflation rises and powerful workers bargain for higher pay, driving up expenses, prices, and wages. This is the fairy-tale narrative of the 1970s, and it's true except that OPEC's embargo drove up oil prices, which produced inflation. Oh well.
Let's be generous to seventies-haunted inflation hawks and say we're worried about a wage-price spiral. Fantastic! No. Real wages are 2.3% lower than they were in Oct 2021 after peaking in June at 4.8%.
Why did America's powerful workers take a paycut rather than demand inflation-based pay? Weak unions, globalization, economic developments.
Workers don't expect inflation to rise, so they're not requesting inflationary hikes. Inflationary expectations have remained moderate, consistent with our data interpretation.
https://www.newyorkfed.org/microeconomics/sce#/
Neither are workers. Working people see surplus savings as wealth and spend it gradually over their lives, despite rising demand. People may have saved money by staying in during the lockdown, but they don't eat out every night to make up for it. Instead, they keep those savings as precautionary balances. This is why the economy is lagging.
People don't buy non-traded goods with pandemic savings (basically, imports). Imports don't multiply like domestic purchases. If you buy a loaf of bread from the corner baker for $1 and they spend it at the tavern across the street, that dollar generates $3 in economic activity. Spending a dollar on foreign goods leaves the country and any multiplier effect happens there, not in the US.
Only marginally higher wages. The ECI is up 1.6% from 2019. Almost all gains went to the 25% lowest-paid Americans. Contrary to the inflation worry about too much savings, these workers don't make enough to save, even post-pandemic.
Recreation and transit spending are at or below pre-pandemic levels. Higher food and hotel prices (which doesn’t mean we’re buying more food than we were in 2019, just that it costs more).
What causes inflation if not greedy workers, free money, and high demand? The most expensive domestic goods produce the biggest revenues for their manufacturers. They charge you more without paying their workers or suppliers more.
The largest price-gougers are funneling their earnings to rich people who store it offshore through stock buybacks and dividends. A $1 billion stock buyback doesn't buy $1 billion in bread.
Five factors influence US inflation today:
I. Price rises for energy and food
II. shifts in consumer tastes
III. supply interruptions (mainly autos);
IV. increased rents (due to telecommuting);
V. monopoly (AKA price-gouging).
None can be remedied by raising interest rates or laying off workers.
Russia's invasion of Ukraine, omicron, and China's Zero Covid policy all disrupted the flow of food, energy, and production inputs. The price went higher because we made less.
After Russia invaded Ukraine, oil prices spiked, and sanctions made it worse. But that was February. By October, oil prices had returned to pre-pandemic, 2015 levels attributable to global economic adjustments, including a shift to renewables. Every new renewable installation reduces oil consumption and affects oil prices.
High food prices have a simple solution. The US and EU have bribed farmers not to produce for 50 years. If the war continues, this program may end, and food prices may decline.
Demand changes. We want different things than in 2019, not more. During the lockdown, people substituted goods. Half of the US toilet-paper supply in 2019 was on commercial-sized rolls. This is created from different mills and stock than our toilet paper.
Lockdown pushed toilet paper demand to residential rolls, causing shortages (the TP hoarding story was just another pandemic urban legend). Because supermarket stores don't have accounts with commercial paper distributors, ordering from languishing stores was difficult. Kleenex and paper towel substitutions caused greater shortages.
All that drove increased costs in numerous product categories, and there were more cases. These increases are transient, caused by supply chain inefficiencies that are resolving.
Demand for frontline staff saw a one-time repricing of pay, which is being recouped as we speak.
Illnesses. Brittle, hollowed-out global supply chains aggravated this. The constant pursuit of cheap labor and minimal regulation by monopolies that dominate most sectors means things are manufactured in far-flung locations. Financialization means any surplus capital assets were sold off years ago, leaving firms with little production slack. After the epidemic, several of these systems took years to restart.
Automobiles are to blame. Financialization and monopolization consolidated microchip and auto production in Taiwan and China. When the lockdowns came, these worldwide corporations cancelled their chip orders, and when they placed fresh orders, they were at the back of the line.
That drove up car prices, which is why the US has slightly higher inflation than other wealthy countries: the economy is car-centric. Automobile prices account for 9% of the CPI. France: 3.6%
Rent shocks and telecommuting. After the epidemic, many professionals moved to exurbs, small towns, and the countryside to work from home. As commercial properties were vacated, it was impractical to adapt them for residential use due to planning restrictions. Addressing these restrictions will cut rent prices more than raising inflation rates, which halts housing construction.
Statistical mirages cause some rent inflation. The CPI estimates what homeowners would pay to rent their properties. When rents rise in your neighborhood, the CPI believes you're spending more on rent even if you have a 30-year fixed-rate mortgage.
Market dominance. Almost every area of the US economy is dominated by monopolies, whose CEOs disclose on investor calls that they use inflation scares to jack up prices and make record profits.
https://pluralistic.net/2022/02/02/its-the-economy-stupid/#overinflated
Long-term profit margins are rising. Markups averaged 26% from 1960-1980. 2021: 72%. Market concentration explains 81% of markup increases (e.g. monopolization). Profit margins reach a 70-year high in 2022. These elements interact. Monopolies thin out their sectors, making them brittle and sensitive to shocks.
If we're worried about a shrinking workforce, there are more humanitarian and sensible solutions than causing a recession and mass unemployment. Instead, we may boost US production capacity by easing workers' entry into the workforce.
https://pluralistic.net/2022/06/01/factories-to-condos-pipeline/#stuff-not-money
US female workforce participation ranks towards the bottom of developed countries. Many women can't afford to work due to America's lack of daycare, low earnings, and bad working conditions in female-dominated fields. If America doesn't have enough workers, childcare subsidies and minimum wages can help.
By contrast, driving the country into recession with interest-rate hikes will reduce employment, and the last recruited (women, minorities) are the first fired and the last to be rehired. Forcing America into recession won't enhance its capacity to create what its people want; it will degrade it permanently.
Nothing the Fed does can stop price hikes from international markets, lack of supply chain investment, COVID-19 disruptions, climate change, the Ukraine war, or market power. They can worsen it. When supply problems generate inflation, raising interest rates decreases investments that can remedy shortages.
Increasing interest rates won't cut rents since landlords pass on the expenses and high rates restrict investment in new dwellings where tenants could escape the costs.
Fixing the supply fixes supply-side inflation. Increase renewables investment (as the Inflation Reduction Act does). Monopolies can be busted (as the IRA does). Reshore key goods (as the CHIPS Act does). Better pay and child care attract employees.
Windfall taxes can claw back price-gouging corporations' monopoly earnings.
https://pluralistic.net/2022/03/15/sanctions-financing/#soak-the-rich
In 2008, we ruled out fiscal solutions (bailouts for debtors) and turned to monetary policy (bank bailouts). This preserved the economy but increased inequality and eroded public trust.
Monetary policy won't help. Even monetary policy enthusiasts recognize an 18-month lag between action and result. That suggests monetary tightening is unnecessary. Like the medieval bloodletter, central bankers whose interest rate hikes don't work swiftly may do more of the same, bringing the economy to its knees.
Interest rates must rise. Zero-percent interest fueled foolish speculation and financialization. Increasing rates will stop this. Increasing interest rates will destroy the economy and dampen inflation.
Then what? All recent evidence indicate to inflation decreasing on its own, as the authors argue. Supply side difficulties are finally being overcome, evidence shows. Energy and food prices are showing considerable mean reversion, which is disinflationary.
The authors don't recommend doing nothing. Best case scenario, they argue, is that the Fed won't keep raising interest rates until morale improves.
Sam Hickmann
3 years ago
Donor-Advised Fund Tax Benefits (DAF)
Giving through a donor-advised fund can be tax-efficient. Using a donor-advised fund can reduce your tax liability while increasing your charitable impact.
Grow Your Donations Tax-Free.
Your DAF's charitable dollars can be invested before being distributed. Your DAF balance can grow with the market. This increases grantmaking funds. The assets of the DAF belong to the charitable sponsor, so you will not be taxed on any growth.
Avoid a Windfall Tax Year.
DAFs can help reduce tax burdens after a windfall like an inheritance, business sale, or strong market returns. Contributions to your DAF are immediately tax deductible, lowering your taxable income. With DAFs, you can effectively pre-fund years of giving with assets from a single high-income event.
Make a contribution to reduce or eliminate capital gains.
One of the most common ways to fund a DAF is by gifting publicly traded securities. Securities held for more than a year can be donated at fair market value and are not subject to capital gains tax. If a donor liquidates assets and then donates the proceeds to their DAF, capital gains tax reduces the amount available for philanthropy. Gifts of appreciated securities, mutual funds, real estate, and other assets are immediately tax deductible up to 30% of Adjusted gross income (AGI), with a five-year carry-forward for gifts that exceed AGI limits.
Using Appreciated Stock as a Gift
Donating appreciated stock directly to a DAF rather than liquidating it and donating the proceeds reduces philanthropists' tax liability by eliminating capital gains tax and lowering marginal income tax.
In the example below, a donor has $100,000 in long-term appreciated stock with a cost basis of $10,000:
Using a DAF would allow this donor to give more to charity while paying less taxes. This strategy often allows donors to give more than 20% more to their favorite causes.
For illustration purposes, this hypothetical example assumes a 35% income tax rate. All realized gains are subject to the federal long-term capital gains tax of 20% and the 3.8% Medicare surtax. No other state taxes are considered.
The information provided here is general and educational in nature. It is not intended to be, nor should it be construed as, legal or tax advice. NPT does not provide legal or tax advice. Furthermore, the content provided here is related to taxation at the federal level only. NPT strongly encourages you to consult with your tax advisor or attorney before making charitable contributions.
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Esteban
3 years ago
The Berkus Startup Valuation Method: What Is It?
What Is That?
Berkus is a pre-revenue valuation method based exclusively on qualitative criteria, like Scorecard.
Few firms match their financial estimates, especially in the early stages, so valuation methodologies like the Berkus method are a good way to establish a valuation when the economic measures are not reliable.
How does it work?
This technique evaluates five key success factors.
Fundamental principle
Technology
Execution
Strategic alliances in its primary market
Production, followed by sales
The Berkus technique values the business idea and four success factors. As seen in the matrix below, each of these dimensions poses a danger to the startup's success.
It assigns $0-$500,000 to each of these beginning regions. This approach enables a maximum $2.5M pre-money valuation.
This approach relies significantly on geography and uses the US as a baseline, as it differs in every country in Europe.
A set of standards for analyzing each dimension individually
Fundamental principle (or strength of the idea)
Ideas are worthless; execution matters. Most of us can relate to seeing a new business open in our area or a startup get funded and thinking, "I had this concept years ago!" Someone did it.
The concept remains. To assess the idea's viability, we must consider several criteria.
The concept's exclusivity It is necessary to protect a product or service's concept using patents and copyrights. Additionally, it must be capable of generating large profits.
Planned growth and growth that goes in a specific direction have a lot of potential, therefore incorporating them into a business is really advantageous.
The ability of a concept to grow A venture's ability to generate scalable revenue is a key factor in its emergence and continuation. A startup needs a scalable idea in order to compete successfully in the market.
The attraction of a business idea to a broad spectrum of people is significantly influenced by the current socio-political climate. Thus, the requirement for the assumption of conformity.
Concept Validation Ideas must go through rigorous testing with a variety of audiences in order to lower risk during the implementation phase.
Technology (Prototype)
This aspect reduces startup's technological risk. How good is the startup prototype when facing cyber threats, GDPR compliance (in Europe), tech stack replication difficulty, etc.?
Execution
Check the management team's efficacy. A potential angel investor must verify the founders' experience and track record with previous ventures. Good leadership is needed to chart a ship's course.
Strategic alliances in its primary market
Existing and new relationships will play a vital role in the development of both B2B and B2C startups. What are the startup's synergies? potential ones?
Production, followed by sales (product rollout)
Startup success depends on its manufacturing and product rollout. It depends on the overall addressable market, the startup's ability to market and sell their product, and their capacity to provide consistent, high-quality support.
Example
We're now founders of EyeCaramba, a machine vision-assisted streaming platform. My imagination always goes to poor puns when naming a startup.
Since we're first-time founders and the Berkus technique depends exclusively on qualitative methods and the evaluator's skill, we ask our angel-investor acquaintance for a pre-money appraisal of EyeCaramba.
Our friend offers us the following table:
Because we're first-time founders, our pal lowered our Execution score. He knows the idea's value and that the gaming industry is red-hot, with worse startup ideas getting funded, therefore he gave the Basic value the highest value (idea).
EyeCaramba's pre-money valuation is $400,000 + $250,000 + $75,000 + $275,000 + $164,000 (1.16M). Good.
References
https://medium.com/humble-ventures/how-angel-investors-value-pre-revenue-startups-part-iii-8271405f0774#:~:text=pre%2Drevenue%20startups.-,Berkus%20Method,potential%20of%20the%20idea%20itself.%E2%80%9D
https://eqvista.com/berkus-valuation-method-for-startups/
https://www.venionaire.com/early-stage-startup-valuation-part-2-the-berkus-method/

Camilla Dudley
3 years ago
How to gain Twitter followers: A 101 Guide
No wonder brands use Twitter to reach their audience. 53% of Twitter users buy new products first.
Twitter growth does more than make your brand look popular. It helps clients trust your business. It boosts your industry standing. It shows clients, prospects, and even competitors you mean business.
How can you naturally gain Twitter followers?
Share useful information
Post visual content
Tweet consistently
Socialize
Spread your @name everywhere.
Use existing customers
Promote followers
Share useful information
Twitter users join conversations and consume material. To build your followers, make sure your material appeals to them and gives value, whether it's sales, product lessons, or current events.
Use Twitter Analytics to learn what your audience likes.
Explore popular topics by utilizing relevant keywords and hashtags. Check out this post on how to use Twitter trends.
Post visual content
97% of Twitter users focus on images, so incorporating media can help your Tweets stand out. Visuals and videos make content more engaging and memorable.
Tweet often
Your audience should expect regular content updates. Plan your ideas and tweet during crucial seasons and events with a content calendar.
Socialize
Twitter connects people. Do more than tweet. Follow industry leaders. Retweet influencers, engage with thought leaders, and reply to mentions and customers to boost engagement.
Micro-influencers can promote your brand or items. They can help you gain new audiences' trust.
Spread your @name everywhere.
Maximize brand exposure. Add a follow button on your website, link to it in your email signature and newsletters, and promote it on business cards or menus.
Use existing customers
Emails can be used to find existing Twitter clients. Upload your email contacts and follow your customers on Twitter to start a dialogue.
Promote followers
Run a followers campaign to boost your organic growth. Followers campaigns promote your account to a particular demographic, and you only pay when someone follows you.
Consider short campaigns to enhance momentum or an always-on campaign to gain new followers.
Increasing your brand's Twitter followers takes effort and experimentation, but the payback is huge.
👋 Follow me on twitter

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