ALBANY — A coalition suing the state to force it to spend more money on New York City public schools may get billions of dollars less than they hoped for when the state’s highest court makes a decision in the 13-year-old case this winter.

At a hearing yesterday at the state Court of Appeals, judges pressed lawyers for the coalition, the Campaign for Fiscal Equity, to explain the rationale behind the $4.7 billion to $5.6 billion increase in annual state operating aid they have argued is necessary for New York City to provide a sound basic public school education. In the past the governor and the legislature have proposed $1.93 billion as an annual minimum increase. Judges grilled campaign lawyers to explain why that amount wouldn’t be enough to satisfy the demands of the lawsuit.

Lawyers defended the $4.7 billion minimum, however, saying it was based on several studies — one commissioned by a lower court — that gave more weight to the poverty rates and cost of living in New York City.

“These are not just artificial figures, this means the difference whether these kids from a poverty background get a fair shake,” a campaign lawyer, Michael Rebell, said during a press conference following the hearing. “We’re talking about a million schoolchildren, 75% who come from poverty backgrounds or disability backgrounds, and you know you can’t get this for cheap.”

It was the third time the case has appeared in front of the Court of Appeals. In 2003, the court upheld a verdict that the state was depriving the New York City public schools of the amount of funding needed to provide the adequate education guaranteed in the state constitution. But the court did not order the state to pay a certain amount, saying that would the breach the separation of powers. Instead, the judges left the exact amount up to the legislature and governor to decide.

They haven’t yet. The court’s deadline was July 2004.In arguments yesterday, lawyers for the plaintiffs asked the court to issue a directive that would force the legislature to allocate a minimum amount.

A lawyer for the state, Denise Hartman, said the state’s pledge of more than $9 billion for school construction and increases in funding to the New York City public schools over the past several years were evidence that the state was already obeying the court order.

“Our position is that we’re moving toward compliance,” she said, adding after further questioning by one of the judges, “The record does not permit us to say how far along we are to compliance.”

There are two new judges on the Court of Appeals since the last time the case appeared there, making the outcome of the case more uncertain. Out of the seven judges on the court, Governor Pataki appointed five and Governor Cuomo appointed two. One judge, Victoria Graffeo, recused herself because she had worked on the lawsuit in the past. If the six judges end up deadlocked, another judge could be chosen by the chief justice, Judith Kaye, to break a tie.

As Governor Pataki left the courthouse yesterday following the swearing in of one of the new judges, Eugene Pigott, Jr., several hundred Campaign for Fiscal Equity supporters surged toward him, chanting for smaller classes and more teachers.

Carrying signs and pictures of school children, many of the marchers were from Albany and other areas outside of New York City — a group whose support the campaign has sought as it tries to expand its base of support. If this case is settled or decided in favor of New York City, it could bring similar claims from other poor urban school districts in the state.

Campaign supporters have put their hopes on the upcoming gubernatorial election if the court doesn’t rule in their favor. The Democratic candidate for governor, Eliot Spitzer, who is leading in the polls, has promised to settle the case within the range of $4.7 to $5.6 billion in additional annual aid. He also has promised not to raise taxes. As attorney general, Mr. Spitzer has also represented the state in its defense against the lawsuit.

New York City now spends about $16 billion a year on its public schools, educating slightly more than a million students. That does not include transportation costs. About $6 billion a year comes from the state.